https://www.math.wisc.edu/wiki/api.php?action=feedcontributions&user=Ckim&feedformat=atomUW-Math Wiki - User contributions [en]2019-09-18T01:20:34ZUser contributionsMediaWiki 1.30.1https://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17883PDE Geometric Analysis seminar2019-09-16T00:14:40Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 28-29, VV901<br />
| Recent progress in analytical aspects of kinetic equations and related fluid models https://www.ki-net.umd.edu/content/conf?event_id=993<br />
|[[# | ]]<br />
| <br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | TBA ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | TBA ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Hyunju Kwon (IAS)<br />
|[[#Hyunju Kwon | TBA ]]<br />
| Kim<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17835PDE Geometric Analysis seminar2019-09-12T19:08:11Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 28-29, VV901<br />
| Recent progress in analytical aspects of kinetic equations and related fluid models https://www.ki-net.umd.edu/content/conf?event_id=993<br />
|[[# | ]]<br />
| <br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | TBA ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | TBA ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17834PDE Geometric Analysis seminar2019-09-12T19:07:44Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 28-29<br />
| Recent progress in analytical aspects of kinetic equations and related fluid models https://www.ki-net.umd.edu/content/conf?event_id=993<br />
|[[# | ]]<br />
| <br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | TBA ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | TBA ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2020-Spring_2021&diff=17679Fall 2020-Spring 20212019-08-26T16:59:06Z<p>Ckim: Created page with "== PDE GA Seminar Schedule Fall 2020-Spring 2021 == {| cellpadding="8" !style="width:20%" align="left" | date !align="left" | speaker !align="left" | title !style="width:..."</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep <br />
| ( )<br />
|[[# | TBA ]]<br />
| <br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17678PDE Geometric Analysis seminar2019-08-26T16:58:16Z<p>Ckim: /* Tentative schedule for Fall 2019-Spring 2020 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | TBA ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | TBA ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17677Fall 2019-Spring 20202019-08-26T16:57:47Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep <br />
| ( )<br />
|[[# | TBA ]]<br />
| <br />
|- <br />
|Sep <br />
| <br />
|[[ # | ]]<br />
| <br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17676PDE Geometric Analysis seminar2019-08-26T16:56:49Z<p>Ckim: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | TBA ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | TBA ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17675PDE Geometric Analysis seminar2019-08-26T16:56:28Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | TBA ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | TBA ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
===Jonathan Luk===<br />
<br />
Title: Stability of vacuum for the Landau equation with moderately soft potentials<br />
<br />
Abstract: Consider the Landau equation with moderately soft potentials in the whole space. We prove that sufficiently small and localized regular initial data give rise to unique global-in-time smooth solutions. Moreover, the solutions approach that of the free transport equation as $t\to +\infty$. This is the first stability of vacuum result for a binary collisional kinetic model featuring a long-range interaction.<br />
<br />
<br />
===Jiaxin Jin===<br />
<br />
Title: Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria.<br />
<br />
Abstract: We first analyze a three-species system with boundary equilibria in some stoichiometric classes and study the rate of convergence to the complex balanced equilibrium. Then we prove similar results on the convergence to the positive equilibrium for a fairly general two-species reversible reaction-diffusion network with boundary equilibria.<br />
<br />
===Jingrui Cheng===<br />
<br />
Title: Gradient estimate for complex Monge-Ampere equations<br />
<br />
Abstract: We consider complex Monge-Ampere equations on a compact Kahler manifold. Previous gradient estimates of the solution all require some derivative bound of the right hand side. I will talk about how to get gradient estimate in $L^p$ and $L^{\infty}$, depending only on the continuity of the right hand side.<br />
<br />
<br />
===Yao Yao===<br />
<br />
Title: Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations<br />
<br />
Abstract: In this talk, I will discuss some recent work on radial symmetry property for stationary or uniformly-rotating solutions for 2D Euler and SQG equation, where we aim to answer the question whether every stationary/uniformly-rotating solution must be radially symmetric, if the vorticity is compactly supported. This is a joint work with Javier Gómez-Serrano, Jaemin Park and Jia Shi.<br />
<br />
===Jessica Lin===<br />
<br />
Title: Speeds and Homogenization for Reaction-Diffusion Equations in Random Media<br />
<br />
Abstract: <br />
The study of spreadings speeds, front speeds, and homogenization for reaction-diffusion equations in random heterogeneous media is of interest for many applications to mathematical modelling. However, most existing arguments rely on the construction of special solutions or linearization techniques. In this talk, I will present some new approaches for their analysis which do not utilize either of these. This talk is based on joint work with Andrej Zlatos.<br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Previous_PDE/GA_seminars&diff=17674Previous PDE/GA seminars2019-08-26T16:55:33Z<p>Ckim: /* Abstracts */</p>
<hr />
<div>[[PDE_Geometric_Analysis_seminar|Click here]] for the current PDE and Geometric Analysis seminar schedule.<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | Stability of vacuum for the Landau equation with moderately soft potentials ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria. ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | Gradient estimate for complex Monge-Ampere equations ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | Speeds and Homogenization for Reaction-Diffusion Equations in Random Media ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Spring 2018 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|January 29, '''3-3:50PM, B341 VV.'''<br />
| Dan Knopf (UT Austin)<br />
|[[#Dan Knopf | Non-K&auml;hler Ricci flow singularities that converge to K&auml;hler-Ricci solitons]]<br />
| Angenent<br />
|- <br />
|February 5, '''3-3:50PM, B341 VV.'''<br />
| Andreas Seeger (UW)<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows ]]<br />
| Kim & Tran<br />
|- <br />
|February 12<br />
| Sam Krupa (UT-Austin)<br />
|[[#Sam Krupa | Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case ]]<br />
| Lee <br />
|- <br />
|February 19<br />
| Maja Taskovic (UPenn)<br />
|[[#Maja Taskovic | Exponential tails for the non-cutoff Boltzmann equation ]]<br />
| Kim<br />
|- <br />
|February 26<br />
| Ashish Kumar Pandey (UIUC)<br />
|[[# | Instabilities in shallow water wave models ]]<br />
| Kim & Lee<br />
|- <br />
|March 5<br />
| Khai Nguyen (NCSU)<br />
|[[#Khai Nguyen | Burgers Equation with Some Nonlocal Sources ]]<br />
| Tran<br />
|- <br />
|March 12<br />
| Hongwei Gao (UCLA)<br />
|[[#Hongwei Gao | Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations ]]<br />
| Tran<br />
|- <br />
|March 19<br />
| Huy Nguyen (Princeton)<br />
|[[#Huy Nguyen | Compressible fluids and active potentials ]]<br />
| Lee<br />
|-<br />
|March 26<br />
| <br />
|[[# | Spring recess (Mar 24-Apr 1, 2018) ]]<br />
| <br />
|-<br />
|April 2<br />
| In-Jee Jeong (Princeton)<br />
|[[#In-Jee Jeong | Singularity formation for the 3D axisymmetric Euler equations ]]<br />
| Kim<br />
|- <br />
|April 9<br />
| Jeff Calder (Minnesota)<br />
|[[#Jeff Calder | Nonlinear PDE continuum limits in data science and machine learning ]]<br />
| Tran<br />
|- <br />
|April 21-22 (Saturday-Sunday)<br />
| [https://sites.google.com/view/81stmidwestpdeseminar/home Midwest PDE seminar]<br />
|[[#Midwest PDE seminar | ]]<br />
| Angenent, Feldman, Kim, Tran.<br />
|- <br />
|April 25 (Wednesday)<br />
| Hitoshi Ishii (Wasow lecture)<br />
|[[#Hitoshi Ishii | Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory]]<br />
| Tran.<br />
|}<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2017 ==<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|September 11<br />
|Mihaela Ifrim (UW)<br />
|[[#Mihaela Ifrim| Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation]]<br />
| Kim & Tran<br />
|- <br />
|September 18<br />
|Longjie Zhang (University of Tokyo) <br />
|[[#Longjie Zhang | On curvature flow with driving force starting as singular initial curve in the plane]]<br />
| Angenent<br />
|- <br />
|September 22,<br />
VV 9th floor hall, 4:00pm<br />
|Jaeyoung Byeon (KAIST) <br />
|[[#Jaeyoung Byeon| Colloquium: Patterns formation for elliptic systems with large interaction forces]]<br />
| Rabinowitz <br />
|- <br />
|September 25<br />
| Tuoc Phan (UTK)<br />
|[[#Tuoc Phan | Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application]]<br />
| Tran<br />
|- <br />
|September 26, <br />
VV B139 4:00pm<br />
| Hiroyoshi Mitake (Hiroshima University)<br />
|[[#Hiroyoshi Mitake | Joint Analysis/PDE seminar: Derivation of multi-layered interface system and its application]]<br />
| Tran<br />
|- <br />
|September 29,<br />
VV901 2:25pm<br />
| Dongnam Ko (CMU & SNU)<br />
|[[#Dongnam Ko | a joint seminar with ACMS: On the emergence of local flocking phenomena in Cucker-Smale ensembles ]]<br />
| Shi Jin & Kim<br />
|- <br />
|October 2<br />
| No seminar due to a KI-Net conference<br />
|<br />
|<br />
|- <br />
|October 9<br />
| Sameer Iyer (Brown University)<br />
|[[#Sameer Iyer | Global-in-x Steady Prandtl Expansion over a Moving Boundary ]]<br />
| Kim<br />
|- <br />
|October 16<br />
| Jingrui Cheng (UW)<br />
|[[#Jingrui Cheng | A 1-D semigeostrophic model with moist convection ]]<br />
| Kim & Tran<br />
|- <br />
|October 23<br />
| Donghyun Lee (UW)<br />
|[[#Donghyun Lee | The Vlasov-Poisson-Boltzmann system in bounded domains ]]<br />
| Kim & Tran<br />
|- <br />
|October 30<br />
| Myoungjean Bae (POSTECH)<br />
|[[#Myoungjean Bae | 3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system ]]<br />
| Feldman<br />
|- <br />
|November 6<br />
| Jingchen Hu (USTC and UW)<br />
|[[#Jingchen Hu | Shock Reflection and Diffraction Problem with Potential Flow Equation ]]<br />
| Kim & Tran <br />
|- <br />
|November 20<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization and Harnack inequality for a degenerate discrete nondivergence form random operator ]]<br />
| Kim & Tran <br />
|- <br />
|November 27 (special time: 3-4PM)<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse problems for Maxwell's equations and its application ]]<br />
| Li<br />
|- <br />
|December 4<br />
| Norbert Pozar (Kanazawa University)<br />
|[[#Norbert Pozar | Viscosity solutions for the crystalline mean curvature flow ]]<br />
| Tran<br />
|- <br />
|December 5 (Tuesday), Colloquium<br />
| Ryan Hynd (U Penn)<br />
|[[#Ryan Hynd | TBD ]]<br />
| <br />
|- <br />
|December 11 (Monday), Colloquium<br />
| Connor Mooney (ETH Zurich)<br />
|[[#Connor Mooney| TBD ]]<br />
| <br />
|}<br />
<br />
<br />
<br />
<br />
= PDE GA Seminar Schedule Spring 2017 =<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|-<br />
|January 23<br>Special time and location:<br> 3-3:50pm, B325 Van Vleck<br />
| Sigurd Angenent (UW)<br />
|[[#Sigurd Angenent | Ancient convex solutions to Mean Curvature Flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|January 30<br />
| Serguei Denissov (UW)<br />
|[[#Serguei Denissov | Instability in 2D Euler equation of incompressible inviscid fluid]]<br />
| Kim & Tran<br />
|- <br />
<br />
<br />
|-<br />
|February 6 - Wasow lecture<br />
| Benoit Perthame (University of Paris VI)<br />
|[[#| ]]<br />
| Jin<br />
|- <br />
<br />
<br />
|-<br />
|February 13<br />
| Bing Wang (UW)<br />
|[[#Bing Wang | The extension problem of the mean curvature flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|February 20<br />
| Eric Baer (UW)<br />
|[[#Eric Baer | Isoperimetric sets inside almost-convex cones]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|February 27<br />
| Ben Seeger (University of Chicago)<br />
|[[#Ben Seeger | Homogenization of pathwise Hamilton-Jacobi equations ]]<br />
| Tran<br />
|- <br />
<br />
|-<br />
|March 7 - Mathematics Department Distinguished Lecture<br />
| Roger Temam (Indiana University) <br />
|[[#Roger Temam | On the mathematical modeling of the humid atmosphere]]<br />
| Smith <br />
|- <br />
<br />
<br />
|-<br />
|March 8 - Analysis/Applied math/PDE seminar<br />
| Roger Temam (Indiana University) <br />
|[[#Roger Temam | Weak solutions of the Shigesada-Kawasaki-Teramoto system ]]<br />
| Smith<br />
|-<br />
<br />
|-<br />
|March 13<br />
| Sona Akopian (UT-Austin)<br />
|[[#Sona Akopian | Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.]]<br />
| Kim<br />
<br />
|-<br />
|March 27 - Analysis/PDE seminar<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | Mean-Field Limits for Ginzburg-Landau vortices ]]<br />
| Tran<br />
<br />
|-<br />
|March 29 - Wasow lecture<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | Microscopic description of Coulomb-type systems ]]<br />
| <br />
<br />
<br />
|-<br />
|March 30 <br>Special day (Thursday) and location:<br> B139 Van Vleck<br />
| Gui-Qiang Chen (Oxford)<br />
|[[#Gui-Qiang Chen | Supersonic Flow onto Solid Wedges,<br />
Multidimensional Shock Waves and Free Boundary Problems ]]<br />
| Feldman<br />
<br />
<br />
<br />
|-<br />
|April 3<br />
| Zhenfu Wang (Maryland)<br />
|[[#Zhenfu Wang | Mean field limit for stochastic particle systems with singular forces]]<br />
| Kim<br />
<br />
|-<br />
|April 10<br />
| Andrei Tarfulea (Chicago)<br />
|[[#Andrei Tarfulea | Improved estimates for thermal fluid equations]]<br />
| Baer<br />
<br />
|-<br />
|April 17 <br>Special time and location:<br> 3-3:50pm, B219 Van Vleck<br />
| Siao-Hao Guo (Rutgers)<br />
|[[# Siao-Hao Guo | Analysis of Velázquez's solution to the mean curvature flow with a type II singularity]]<br />
| Lu Wang<br />
<br />
<br />
|-<br />
|April 24<br />
| Jianfeng Lu (Duke)<br />
|[[#Jianfeng Lu | Evolution of crystal surfaces: from mesoscopic to continuum models]]<br />
| Li<br />
<br />
|-<br />
|April 25- joint Analysis/PDE seminar<br />
| Chris Henderson (Chicago)<br />
|[[#Chris Henderson | A local-in-time Harnack inequality and applications to reaction-diffusion equations<br />
<br />
]]<br />
| Lin<br />
<br />
|-<br />
|May 1st(Special time: 4:00-5:00pm)<br />
| Jeffrey Streets (UC-Irvine)<br />
|[[#Jeffrey Streets | Generalized Kahler Ricci flow and a generalized Calabi conjecture]]<br />
| Bing Wang<br />
|}<br />
<br />
<br />
<br />
= PDE GA Seminar Schedule Fall 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 12<br />
| Daniel Spirn (U of Minnesota)<br />
|[[#Daniel Spirn | Dipole Trajectories in Bose-Einstein Condensates ]]<br />
| Kim<br />
|-<br />
|September 19<br />
| Donghyun Lee (UW-Madison)<br />
|[[#Donghyun Lee | The Boltzmann equation with specular boundary condition in convex domains ]]<br />
| Feldman<br />
|-<br />
|September 26<br />
| Kevin Zumbrun (Indiana)<br />
|[[#Kevin Zumbrun | A Stable Manifold Theorem for a class of degenerate evolution equations ]]<br />
| Kim <br />
|-<br />
|October 3<br />
| Will Feldman (UChicago )<br />
|[[#Will Feldman | Liquid Drops on a Rough Surface ]]<br />
| Lin & Tran<br />
|-<br />
|October 10<br />
| Ryan Hynd (UPenn)<br />
|[[#Ryan Hynd | Extremal functions for Morrey’s inequality in convex domains ]]<br />
| Feldman<br />
|-<br />
|October 17<br />
| Gung-Min Gie (Louisville)<br />
|[[#Gung-Min Gie | Boundary layer analysis of some incompressible flows ]]<br />
| Kim<br />
|-<br />
|October 24<br />
| Tau Shean Lim (UW Madison)<br />
|[[#Tau Shean Lim | Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators ]]<br />
| Kim & Tran<br />
|-<br />
|October 31 ('''Special time and room''': B313VV, 3PM-4PM)<br />
| Tarek Elgindi ( Princeton)<br />
|[[#Tarek Elgindi | Propagation of Singularities in Incompressible Fluids ]]<br />
| Lee & Kim<br />
|-<br />
|November 7<br />
| Adrian Tudorascu (West Virginia)<br />
|[[#Adrian Tudorascu | Hamilton-Jacobi equations in the Wasserstein space of probability measures ]]<br />
| Feldman<br />
|-<br />
|November 14<br />
| Alexis Vasseur ( UT-Austin)<br />
|[[#Alexis Vasseur | Compressible Navier-Stokes equations with degenerate viscosities ]]<br />
| Feldman<br />
|-<br />
|November 21<br />
| Minh-Binh Tran (UW Madison )<br />
|[[#Minh-Binh Tran | Quantum Kinetic Problems ]]<br />
| Hung Tran<br />
|-<br />
|November 28<br />
| David Kaspar (Brown)<br />
|[[#David Kaspar | Kinetics of shock clustering ]]<br />
|Tran<br />
|-<br />
|December 5 ('''Special time and room''': 3PM-4PM, B313VV)<br />
| Brian Weber (University of Pennsylvania)<br />
|[[#Brian Weber | Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds ]]<br />
|Bing Wang<br />
|-<br />
|December 12<br />
| (no seminar)<br />
|[[# | ]]<br />
| <br />
|}<br />
<br />
<br />
<br />
= Seminar Schedule Spring 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 25<br />
||Tianling Jin (HKUST and Caltech)<br />
|[[#Tianling Jin | Holder gradient estimates for parabolic homogeneous p-Laplacian equations ]]<br />
| Zlatos<br />
|-<br />
|February 1<br />
|Russell Schwab (Michigan State University)<br />
|[[#Russell Schwab | Neumann homogenization via integro-differential methods ]]<br />
| Lin<br />
|-<br />
|February 8 <br />
|Jingrui Cheng (UW Madison)<br />
|[[#Jingrui Cheng | Semi-geostrophic system with variable Coriolis parameter ]]<br />
| Tran & Kim<br />
|-<br />
|February 15 <br />
|Paul Rabinowitz (UW Madison)<br />
|[[#Paul Rabinowitz | On A Double Well Potential System ]]<br />
| Tran & Kim<br />
|-<br />
|February 22 <br />
|Hong Zhang (Brown)<br />
|[[#Hong Zhang | On an elliptic equation arising from composite material ]]<br />
| Kim<br />
|-<br />
|February 29<br />
|Aaron Yip (Purdue university) <br />
|[[#Aaron Yip | Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media ]]<br />
| Tran<br />
|-<br />
|March 7<br />
|Hiroyoshi Mitake (Hiroshima university) <br />
||[[#Hiroyoshi Mitake | Selection problem for fully nonlinear equations]]<br />
| Tran<br />
|-<br />
|March 15<br />
|Nestor Guillen (UMass Amherst)<br />
|[[#Nestor Guillen | Min-max formulas for integro-differential equations and applications ]]<br />
| Lin<br />
|-<br />
|March 21 (Spring Break)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 28<br />
|Ryan Denlinger (Courant Institute)<br />
|[[#Ryan Denlinger | The propagation of chaos for a rarefied gas of hard spheres in vacuum ]]<br />
| Lee<br />
|-<br />
|April 4<br />
| No seminar<br />
||[[# | ]]<br />
| <br />
|-<br />
|April 11<br />
|Misha Feldman (UW)<br />
|[[#Misha Feldman | Shock reflection, free boundary problems and degenerate elliptic equations ]]<br />
|<br />
|-<br />
|April 14: 2:25 PM in VV 901-Joint with Probability Seminar<br />
|Jessica Lin (UW-Madison)<br />
|[[#Jessica Lin | Optimal Quantitative Error Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form ]]<br />
|-<br />
|April 18<br />
|Sergey Bolotin (UW-Madison)<br />
|[[#Sergey Bolotin | Degenerate billiards]]<br />
|<br />
|-<br />
|April 21-24, KI-Net conference: Boundary Value Problems and Multiscale Coupling Methods for Kinetic Equations<br />
|Link: http://www.ki-net.umd.edu/content/conf?event_id=493<br />
|-<br />
|April 25<br />
| Moon-Jin Kang (UT-Austin)<br />
|[[#Moon-Jin Kang | On contraction of large perturbation of shock waves, and inviscid limit problems ]]<br />
| Kim<br />
|-<br />
|Tuesday, May 3, 4:00 p.m., in B139 (Joint Analysis-PDE seminar )<br />
|Stanley Snelson (University of Chicago) <br />
|[[# | ]]<br />
| Seeger & Tran.<br />
|-<br />
|May 16-20, Conference in Harmonic Analysis in Honor of Michael Christ<br />
|Link: https://www.math.wisc.edu/ha_2016/<br />
|-<br />
|}<br />
<br />
<br />
<br />
= Seminar Schedule Fall 2015 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|September 7 (Labor Day)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|September 14 (special room: B115)<br />
| Hung Tran (Madison)<br />
|[[#Hung Tran | Some inverse problems in periodic homogenization of Hamilton--Jacobi equations ]]<br />
| <br />
|- <br />
|September 21 (special room: B115) <br />
| Eric Baer (Madison)<br />
||[[#Eric Baer | Optimal function spaces for continuity of the Hessian determinant as a distribution ]]<br />
| <br />
|-<br />
|September 28<br />
| Donghyun Lee (Madison)<br />
|[[#Donghyun Lee | FLUIDS WITH FREE-SURFACE AND VANISHING VISCOSITY LIMIT]]<br />
| <br />
|-<br />
|October 5 <br />
|Hyung-Ju Hwang (Postech & Brown Univ)<br />
|[[#Hyung-Ju Hwang | The Fokker-Planck equation in bounded domains ]]<br />
| Kim<br />
|-<br />
|October 12<br />
| Minh-Binh Tran (Madison)<br />
|[[#Minh-Binh Tran | Nonlinear approximation theory for kinetic equations ]]<br />
| <br />
|-<br />
|October 19<br />
| Bob Jensen (Loyola University Chicago)<br />
||[[#Bob Jensen | Crandall-Lions Viscosity Solutions of Uniformly Elliptic PDEs ]]<br />
| Tran<br />
|-<br />
|October 26<br />
|Luis Silvestre (Chicago)<br />
|[[#Luis Silvestre | A priori estimates for integral equations and the Boltzmann equation ]]<br />
|Kim<br />
|-<br />
|November 2<br />
| Connor Mooney (UT Austin)<br />
|[[#Connor Mooney | Counterexamples to Sobolev regularity for degenerate Monge-Ampere equations ]]<br />
|Lin<br />
|-<br />
|November 9<br />
| Javier Gomez Serrano (Princeton)<br />
||[[#Javier Gomez Serrano | Existence and regularity of rotating global solutions for active scalars ]]<br />
|Zlatos<br />
|-<br />
|November 16<br />
| Yifeng Yu (UC Irvine)<br />
|[[#Yifeng Yu | G-equation in the modeling of flame propagation ]]<br />
| Tran<br />
|-<br />
|November 23<br />
| Nam Le (Indiana)<br />
|[[#Nam Le | Global smoothness of the Monge-Ampere eigenfunctions ]]<br />
|Tran<br />
|-<br />
|November 30<br />
| Qin Li (Madison)<br />
|[[#Qin Li | Kinetic-fluid coupling: transition from the Boltzmann to the Euler ]]<br />
|<br />
|-<br />
|December 7<br />
| Lu Wang (Madison) <br />
||[[#Lu Wang | Asymptotic Geometry of Self-shrinkers ]]<br />
|<br />
|-<br />
|December 14<br />
| Christophe Lacave (Paris 7)<br />
|[[# Christophe Lacave | Well-posedness for 2D Euler in non-smooth domains ]]<br />
| Zlatos<br />
|}<br />
<br />
<br />
<br />
= Seminar Schedule Spring 2015 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 21 (Departmental Colloquium: 4PM, B239) <br />
|Jun Kitagawa (Toronto) <br />
|[[#Jun Kitagawa (Toronto) | Regularity theory for generated Jacobian equations: from optimal transport to geometric optics ]]<br />
|Feldman <br />
|-<br />
|February 9 <br />
|Jessica Lin (Madison)<br />
|[[#Jessica Lin (Madison) | Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations ]]<br />
|Kim<br />
|-<br />
|February 17 (Tuesday) (joint with Analysis Seminar: 4PM, B139)<br />
|Chanwoo Kim (Madison) <br />
|[[#Chanwoo Kim (Madison) | Hydrodynamic limit from the Boltzmann to the Navier-Stokes-Fourier ]]<br />
|Seeger <br />
|-<br />
|February 23 (special time*, '''3PM, B119''') <br />
| Yaguang Wang (Shanghai Jiao Tong)<br />
|[[ #Yaguang Wang | Stability of Three-dimensional Prandtl Boundary Layers ]]<br />
|Jin<br />
|-<br />
|March 2 <br />
|Benoit Pausader (Princeton)<br />
|[[#Benoit Pausader (Princeton) | Global smooth solutions for the Euler-Maxwell problem for electrons in 2 dimensions]]<br />
|Kim<br />
|-<br />
|March 9 <br />
|Haozhao Li (University of Science and Technology of China) <br />
|[[#Haozhao Li|Regularity scales and convergence of the Calabi flow]]<br />
|Wang <br />
|-<br />
|March 16 <br />
| Jennifer Beichman (Madison) <br />
|[[#Jennifer Beichman (Madison) |Nonstandard dispersive estimates and linearized water waves ]]<br />
| Kim<br />
|-<br />
|March 23 <br />
| Ben Fehrman (University of Chicago)<br />
|[[#Ben Fehrman (University of Chicago) | On The Existence of an Invariant Measure for Isotropic Diffusions in Random Environments ]]<br />
| Lin<br />
|-<br />
|March 30 <br />
| Spring recess Mar 28-Apr 5 (S-N)<br />
|[[# | ]]<br />
| <br />
|-<br />
|April 13 <br />
| Sung-Jin Oh (Berkeley)<br />
|[[# Berkeley | Global well-posedness of the energy critical Maxwell-Klein-Gordon equation ]]<br />
| Kim<br />
|-<br />
|April 20 <br />
|Yuan Lou (Ohio State)<br />
|[[#Yuan Lou (Ohio State) | Asymptotic behavior of the smallest eigenvalue of an elliptic operator and its applications to evolution of dispersal]]<br />
|Zlatos<br />
|-<br />
|'''April 28''' (a joint seminar with analysis, '''4:00 p.m B139''')<br />
| Diego Córdoba (ICMAT, Madrid) <br />
|[[# Diego Córdoba |Global existence solutions and geometric properties of the SQG sharp front ]]<br />
| Zlatos<br />
|-<br />
|May 4 <br />
| Vera Hur (UIUC) <br />
|[[# Vera Hur (UIUC) |Instabilities in nonlinear dispersive waves ]]<br />
| Yao<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Fall 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker <br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 15 <br />
|Greg Kuperberg (UC-Davis)<br />
|[[#Greg Kuperberg| ''Cartan-Hadamard and the Little Prince'' ]]<br />
|Viaclovsky<br />
|-<br />
|-<br />
|September 22 (joint with Analysis Seminar)<br />
|Steve Hofmann (U. of Missouri)<br />
|[[#Steve Hofmann| ''Quantitative Rectifiability and Elliptic Equations'']]<br />
|Seeger<br />
|-<br />
|-<br />
|Oct 6th, <br />
|Xiangwen Zhang (Columbia University)<br />
|[[#Xiangwen Zhang |Alexandrov's Uniqueness Theorem for Convex Surfaces ]]<br />
|B.Wang<br />
|-<br />
|-<br />
|October 13 <br />
|Xuwen Chen (Brown University)[http://www.math.brown.edu/~chenxuwen/]<br />
|[[#Xuwen Chen| The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-body Evolution]]<br />
|C.Kim<br />
|-<br />
|-<br />
|October 20 <br />
|Kyudong Choi (UW-Madison)<br />
|[[#Kyudong Choi| Finite time blow up for 1D models for the 3D Axisymmetric Euler Equations the 2D Boussinesq system]]<br />
|C.Kim<br />
|-<br />
|-<br />
|October 27<br />
|Chanwoo Kim (UW-Madison)<br />
|[[#Chanwoo Kim|<br />
BV-Regularity of the Boltzmann Equation in Non-Convex Domains]]<br />
|Local<br />
|-<br />
|-<br />
|November 3 <br />
|Myoungjean Bae (POSTECH)<br />
|[[#Myoungjean Bae| Recent progress on study of Euler-Poisson system ]]<br />
|M.Feldman<br />
|-<br />
|-<br />
|November 10 <br />
|Philip Isett (MIT)<br />
|[[#Philip Isett| Hölder Continuous Euler Flows ]]<br />
|C.Kim<br />
|-<br />
|-<br />
|November 17<br />
|Lei Wu<br />
|[[#Lei Wu | Geometric Correction for Diffusive Expansion in Neutron Transport Equation ]]<br />
|C.Kim<br />
|-<br />
|-<br />
|December 1<br />
|[http://orion.math.iastate.edu/xhnguyen Xuan Hien Nguyen] (Iowa State University)<br />
|[[#Xuan Hien Nguyen| Gluing constructions for self-similar surfaces under mean curvature flow ]]<br />
|Angenent<br />
|-<br />
|}<br />
<br />
<br />
<br />
<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 14 at 4pm in B139 (TUESDAY), joint with Analysis<br />
|[http://www.math.univ-toulouse.fr/~roque/ Jean-Michel Roquejoffre (Toulouse)]<br />
|[[#Jean-Michel Roquejoffre (Toulouse) |<br />
Front propagation in the presence of integral diffusion. ]]<br />
|Zlatos<br />
|-<br />
|-<br />
|February 10<br />
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]<br />
|[[#Myoungjean Bae (POSTECH) |<br />
Free Boundary Problem related to Euler-Poisson system. ]]<br />
|Feldman<br />
|-<br />
|-<br />
|February 24<br />
|[http://www2.math.umd.edu/~jefftan/ Changhui Tan (Maryland)]<br />
|[[#Changhui Tan (Maryland) |<br />
Global classical solution and long time behavior of macroscopic flocking models. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
Parabolic equations in time-varying domains. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 10<br />
|[http://math.uchicago.edu/~jiahao/ Hao Jia (University of Chicago)]<br />
|[[#Hao Jia (University of Chicago) |<br />
Long time dynamics of energy critical defocusing wave equation with<br />
radial potential in 3+1 dimensions. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 31<br />
|[http://www.mth.kcl.ac.uk/staff/a_pushnitski.html Alexander Pushnitski (King's College London)]<br />
|[[#Alexander Pushnitski (King's College) |<br />
An inverse spectral problem for Hankel operators. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 21<br />
|[http://people.math.gatech.edu/~panrh/ Ronghua Pan (Georgia Tech)]<br />
|[[#Ronghua Pan (Georgia Tech) |<br />
Compressible Navier-Stokes-Fourier system with temperature dependent dissipation. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 9<br />
|Greg Drugan (U. of Washington)<br />
|[[#Greg Drugan (U. of Washington) |<br />
Construction of immersed self-shrinkers]]<br />
|Angenent<br />
|-<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 25<br />
|Myeongju Chae (Hankyong National University visiting UW)<br />
|[[#Myeongju Chae (Hankyong National University) |<br />
On the global classical solution of the Keller-Segel-Navier -Stokes system and its asymptotic behavior. ]]<br />
|Kiselev<br />
|-<br />
myeongju Chae <br />
|-<br />
|December 2<br />
|Xiaojie Wang<br />
|[[#Xiaojie Wang (Stony Brook University) |<br />
Uniqueness of Ricci flow solutions on noncompact manifolds. ]]<br />
|Wang<br />
|-<br />
|-<br />
|December 16<br />
|Antonio Ache(Princeton)<br />
|[[#Antonio Ache(Princeton) |<br />
Ricci Curvature and the manifold learning problem. NOTE: Due to final exams, this seminar will be held in B231. ]]<br />
|Viaclovsky<br />
|-<br />
|}<br />
<br />
<br />
<br />
= Seminar Schedule Spring 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|-<br />
|February 4<br />
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]<br />
|[[#Myoungjean Bae (POSTECH) |<br />
Transonic shocks for Euler-Poisson system and related problems]]<br />
|Feldman<br />
|-<br />
|-<br />
|February 18<br />
|Mike Cullen (Met. Office, UK)<br />
|[[ #Mike Cullen (Met. Office, UK)|<br />
Modelling the uncertainty in predicting large-scale atmospheric circulations.]]<br />
|Feldman<br />
|-<br />
|-<br />
|March 18<br />
|Mohammad Ghomi(Math. Georgia Tech)<br />
|[[ #Mohammad Ghomi(Math. Georgia Tech)|<br />
Tangent lines, inflections, and vertices of closed curves.]]<br />
|Angenent<br />
|-<br />
|-<br />
|April 8<br />
|[http://www.maths.ox.ac.uk/contact/details/xiang Wei Xiang (Oxford)]<br />
|[[ #Wei Xiang (Oxford)|<br />
Shock Diffraction Problem to the<br />
Two Dimensional Nonlinear Wave System and Potential Flow Equation.]]<br />
|Feldman<br />
|-<br />
|-<br />
|Thursday April 25, Time: 3:30, Room: 901 Van Vleck (Note: Special Day, Room)<br />
|[http://math.wvu.edu/~adriant/CV1.html Adrian Tudorascu (West Virginia University)]<br />
|[[ #Adrian Tudorascu (West Virginia University)|One-dimensional pressureless <br />
Euler/Euler-Poisson systems with/without viscosity<br />
.]]<br />
|Feldman<br />
|-<br />
|-<br />
|May 6<br />
|[http://www.icmat.es/dcordoba Diego Cordoba (Madrid)]<br />
|[[#Diego Cordoba (Madrid) |<br />
Interface dynamics for incompressible fluids. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Fall 2012 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 17<br />
|[http://www.math.wisc.edu/~bwang/ Bing Wang (UW Madison)]<br />
|[[#Bing Wang (UW Madison)|<br />
On the regularity of limit space]]<br />
|local<br />
|-<br />
|-<br />
|October 15<br />
|[http://www.math.umn.edu/~polacik/ Peter Polacik (University of Minnesota)]<br />
|[[#Peter Polacik (University of Minnesota)|<br />
Exponential separation between positive and sign-changing solutions and its applications]]<br />
|Zlatos<br />
|-<br />
|-<br />
|November 26<br />
|[http://www.math.wisc.edu/~kchoi/ Kyudong Choi (UW Madison)]<br />
|[[#Kyudong Choi (UW Madison)|<br />
Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations]]<br />
|local<br />
|-<br />
|-<br />
|December 10<br />
|[http://www.math.wisc.edu/~yaoyao/ Yao Yao (UW Madison)]<br />
|[[#Yao Yao (UW Madison)|<br />
Confinement for nonlocal interaction equation with repulsive-attractive kernels]]<br />
|local<br />
|-<br />
|}<br />
<br />
<br />
= Seminar Schedule Spring 2012 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Feb 6<br />
|Yao Yao (UCLA)<br />
|[[#Yao Yao (UCLA)|<br />
Degenerate diffusion with nonlocal aggregation: behavior of solutions]]<br />
|Kiselev<br />
|-<br />
|March 12<br />
| Xuan Hien Nguyen (Iowa State)<br />
|[[#Xuan Hien Nguyen (Iowa State)|<br />
Gluing constructions for solitons and self-shrinkers under mean curvature flow]]<br />
|Angenent<br />
|-<br />
|March 21(Wednesday!), Room 901 Van Vleck<br />
|Nestor Guillen (UCLA)<br />
|[[#Nestor Guillen (UCLA)|<br />
The local geometry of maps with c-convex potentials]]<br />
|Feldman<br />
|-<br />
|March 26<br />
|Vlad Vicol (University of Chicago)<br />
|[[#Vlad Vicol (U Chicago)|<br />
Shape dependent maximum principles and applications]]<br />
|Kiselev<br />
|-<br />
|April 9<br />
|Charles Smart (MIT) <br />
|[[#Charles Smart (MIT)|<br />
PDE methods for the Abelian sandpile<br />
]]<br />
|Seeger<br />
|-<br />
|April 16<br />
|Jiahong Wu (Oklahoma)<br />
|[[#Jiahong Wu (Oklahoma State)|<br />
The 2D Boussinesq equations with partial dissipation]]<br />
|Kiselev<br />
|-<br />
|April 23<br />
|Joana Oliveira dos Santos Amorim (Universite Paris Dauphine)<br />
|[[#Joana Oliveira dos Santos Amorim (Universite Paris Dauphine)|<br />
A geometric look on Aubry-Mather theory and a theorem of Birkhoff]]<br />
|Bolotin<br />
|-<br />
|April 27 (Colloquium. Friday at 4pm, in Van Vleck B239)<br />
|Gui-Qiang Chen (Oxford) <br />
|[[#Gui-Qiang Chen (Oxford) |<br />
Nonlinear Partial Differential Equations of Mixed Type ]]<br />
|Feldman<br />
|-<br />
|May 14<br />
|Jacob Glenn-Levin (UT Austin)<br />
|[[#Jacob Glenn-Levin (UT Austin)|<br />
Incompressible Boussinesq equations in borderline Besov spaces]]<br />
|Kiselev<br />
|}<br />
<br />
<br />
= Seminar Schedule Fall 2011 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Oct 3<br />
|Takis Souganidis (Chicago)<br />
|[[#Takis Souganidis (Chicago)|<br />
''Stochastic homogenization of the G-equation'']]<br />
|Armstrong<br />
<br />
|-<br />
|Oct 10<br />
|Scott Armstrong (UW-Madison)<br />
|[[#Scott Armstrong (UW-Madison)|<br />
''Partial regularity for fully nonlinear elliptic equations'']]<br />
|Local speaker<br />
<br />
|-<br />
|Oct 17<br />
|Russell Schwab (Carnegie Mellon)<br />
|[[#Russell Schwab (Carnegie Mellon)|<br />
''On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)'']]<br />
|Armstrong<br />
|-<br />
|October 24 ( with Geometry/Topology seminar)<br />
|[http://math.univ-lyon1.fr/~ovsienko/ Valentin Ovsienko] (University of Lyon)<br />
|[[#Valentin Ovsienko (University of Lyon)|<br />
''The pentagram map and generalized friezes of Coxeter'']]<br />
|Marí Beffa<br />
|-<br />
|-<br />
|Oct 31<br />
|Adrian Tudorascu (West Virginia University)<br />
|[[#Adrian Tudorascu (West Virginia University)|<br />
''Weak Lagrangian solutions for the Semi-Geostrophic system in physical space'']]<br />
|Feldman<br />
|-<br />
|Nov 7<br />
|James Nolen (Duke)<br />
|[[#James Nolen (Duke)|<br />
''Normal approximation for a random elliptic PDE'']]<br />
|Armstrong<br />
|-|-<br />
|Nov 21 (Joint with [http://www.math.wisc.edu/~seeger/curr.html Analysis] seminar)<br />
|Betsy Stovall (UCLA)<br />
|[[#Betsy Stovall (UCLA)| <br />
''Scattering for the cubic Klein--Gordon equation in two dimensions'']]<br />
|Seeger<br />
|-<br />
|Dec 5<br />
|Charles Smart (MIT)<br />
|[[#Charles Smart (MIT)|<br />
''Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian'']]<br />
|Armstrong<br />
|}<br />
<br />
= Seminar Schedule Spring 2011 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Jan 24<br />
|Bing Wang (Princeton)<br />
|[[#Bing Wang (Princeton)|<br />
''The Kaehler Ricci flow on Fano manifold '']]<br />
|Viaclovsky<br />
|-<br />
|Mar 15 (TUESDAY) at 4pm in B139 (joint wit Analysis)<br />
|Francois Hamel (Marseille)<br />
|[[#Francois Hamel (Marseille)|<br />
''Optimization of eigenvalues of non-symmetric elliptic operators'']]<br />
|Zlatos<br />
|-<br />
|Mar 28<br />
|Juraj Foldes (Vanderbilt)<br />
|[[#Juraj Foldes (Vanderbilt)|<br />
''Symmetry properties of parabolic problems and their applications'']]<br />
|Zlatos<br />
|-<br />
|Apr 11<br />
|Alexey Cheskidov (UIC)<br />
|[[#Alexey Cheskidov (UIC)|<br />
''Navier-Stokes and Euler equations: a unified approach to the problem of blow-up'']]<br />
|Kiselev<br />
|-<br />
|Date TBA<br />
|Mikhail Feldman (UW Madison)<br />
|''TBA''<br />
|Local speaker<br />
|-<br />
|Date TBA<br />
|Sigurd Angenent (UW Madison)<br />
|''TBA''<br />
|Local speaker<br />
|-<br />
|}<br />
= Seminar Schedule Fall 2010 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 13<br />
|Fausto Ferrari (Bologna)<br />
|[[#Fausto Ferrari (Bologna)|<br />
''Semilinear PDEs and some symmetry properties of stable solutions'']]<br />
|Feldman<br />
|-<br />
|Sept 27<br />
|Arshak Petrosyan (Purdue)<br />
|[[#Arshak Petrosyan (Purdue)|<br />
''Nonuniqueness in a free boundary problem from combustion'']]<br />
|Feldman<br />
|-<br />
|Oct 7, Thursday, 4:30 pm, Room: 901 Van Vleck. '''Special day, time & room.'''<br />
|Changyou Wang (U. of Kentucky)<br />
|[[#Changyou Wang (U. of Kentucky)|<br />
''Phase transition for higher dimensional wells'']]<br />
|Feldman<br />
|-<br />
|Oct 11<br />
|Philippe LeFloch (Paris VI)<br />
|[[#Philippe LeFloch (Paris VI)|<br />
''Kinetic relations for undercompressive shock waves and propagating phase boundaries'']]<br />
|Feldman<br />
|-<br />
|Oct 29 Friday 2:30pm, Room: B115 Van Vleck. '''Special day, time & room.'''<br />
|[http://www.ima.umn.edu/~imitrea/ Irina Mitrea] (IMA)<br />
|[[#Irina Mitrea |<br />
''Boundary Value Problems for Higher Order Differential Operators'']]<br />
|[https://www.math.wisc.edu/~wimaw/ WiMaW]<br />
|-<br />
|-<br />
|Nov 1<br />
|Panagiota Daskalopoulos (Columbia U)<br />
|[[#Panagiota Daskalopoulos (Columbia U)|<br />
''Ancient solutions to geometric flows'']]<br />
|Feldman<br />
|-<br />
|Nov 8<br />
|Maria Gualdani (UT Austin)<br />
|[[#Maria Gualdani (UT Austin)|<br />
''A nonlinear diffusion model in mean-field games'']]<br />
|Feldman<br />
|-<br />
|Nov 18 Thursday 1:20pm Room: 901 Van Vleck '''Special day & time.'''<br />
|Hiroshi Matano (Tokyo University) <br />
|[[#Hiroshi Matano (Tokyo University)|<br />
''Traveling waves in a sawtoothed cylinder and their homogenization limit'']]<br />
|Angenent & Rabinowitz<br />
|-<br />
|Nov 29<br />
|Ian Tice (Brown University)<br />
|[[#Ian Tice (Brown University)|<br />
''Global well-posedness and decay for the viscous surface wave<br />
problem without surface tension'']]<br />
|Feldman<br />
|-<br />
|Dec. 8 Wed 2:25pm, Room: 901 Van Vleck. '''Special day, time & room.'''<br />
|Hoai Minh Nguyen (NYU-Courant Institute)<br />
|[[#Hoai Minh Nguyen (NYU-Courant Institute)|<br />
''Cloaking via change of variables for the Helmholtz equation'']]<br />
|Feldman<br />
|-<br />
|}<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
==Abstracts==<br />
<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
===Jonathan Luk===<br />
<br />
Title: Stability of vacuum for the Landau equation with moderately soft potentials<br />
<br />
Abstract: Consider the Landau equation with moderately soft potentials in the whole space. We prove that sufficiently small and localized regular initial data give rise to unique global-in-time smooth solutions. Moreover, the solutions approach that of the free transport equation as $t\to +\infty$. This is the first stability of vacuum result for a binary collisional kinetic model featuring a long-range interaction.<br />
<br />
<br />
===Jiaxin Jin===<br />
<br />
Title: Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria.<br />
<br />
Abstract: We first analyze a three-species system with boundary equilibria in some stoichiometric classes and study the rate of convergence to the complex balanced equilibrium. Then we prove similar results on the convergence to the positive equilibrium for a fairly general two-species reversible reaction-diffusion network with boundary equilibria.<br />
<br />
===Jingrui Cheng===<br />
<br />
Title: Gradient estimate for complex Monge-Ampere equations<br />
<br />
Abstract: We consider complex Monge-Ampere equations on a compact Kahler manifold. Previous gradient estimates of the solution all require some derivative bound of the right hand side. I will talk about how to get gradient estimate in $L^p$ and $L^{\infty}$, depending only on the continuity of the right hand side.<br />
<br />
<br />
===Yao Yao===<br />
<br />
Title: Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations<br />
<br />
Abstract: In this talk, I will discuss some recent work on radial symmetry property for stationary or uniformly-rotating solutions for 2D Euler and SQG equation, where we aim to answer the question whether every stationary/uniformly-rotating solution must be radially symmetric, if the vorticity is compactly supported. This is a joint work with Javier Gómez-Serrano, Jaemin Park and Jia Shi.<br />
<br />
===Jessica Lin===<br />
<br />
Title: Speeds and Homogenization for Reaction-Diffusion Equations in Random Media<br />
<br />
Abstract: <br />
The study of spreadings speeds, front speeds, and homogenization for reaction-diffusion equations in random heterogeneous media is of interest for many applications to mathematical modelling. However, most existing arguments rely on the construction of special solutions or linearization techniques. In this talk, I will present some new approaches for their analysis which do not utilize either of these. This talk is based on joint work with Andrej Zlatos.<br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.<br />
<br />
<br />
===Dan Knopf===<br />
<br />
Title: Non-K&auml;hler Ricci flow singularities that converge to K&auml;hler-Ricci solitons<br />
<br />
Abstract: We describe Riemannian (non-K&auml;hler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking K&auml;hler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered by Feldman, Ilmanen, and Knopf in 2003. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-K&auml;hler solutions of Ricci flow that become asymptotically K&auml;hler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of K&auml;hler metrics under Ricci flow.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Sam Krupa===<br />
<br />
Title: Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case<br />
<br />
Abstract: For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (Panov). This single entropy result was proven again by De Lellis, Otto and Westdickenberg in 2004. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In our new work, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case. This is joint work with A. Vasseur.<br />
<br />
<br />
===Maja Taskovic===<br />
<br />
Title: Exponential tails for the non-cutoff Boltzmann equation<br />
<br />
Abstract: The Boltzmann equation models the motion of a rarefied gas, in which particles interact through binary collisions, by describing the evolution of the particle density function. The effect of collisions on the density function is modeled by a bilinear integral operator (collision operator) which in many cases has a non-integrable angular kernel. For a long time the equation was simplified by assuming that this kernel is integrable (the so called Grad's cutoff) with a belief that such an assumption does not affect the equation significantly. However, in the last 20 years it has been observed that a non-integrable singularity carries regularizing properties which motivates further analysis of the equation in this setting.<br />
<br />
We study behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime by examining the generation and propagation of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. For this purpose we introduce Mittag-Leffler moments which can be understood as a generalization of exponential moments. An interesting aspect of this result is that the singularity rate of the angular kernel affects the order of tails that can be shown to propagate in time. This is based on joint works with Alonso, Gamba, Pavlovic and Gamba, Pavlovic.<br />
<br />
<br />
===Ashish Kumar Pandey===<br />
<br />
Title: Instabilities in shallow water wave models<br />
<br />
Abstract: Slow modulations in wave characteristics of a nonlinear, periodic traveling wave in a dispersive medium may develop non-trivial structures which evolve as it propagates. This phenomenon is called modulational instability. In the context of water waves, this phenomenon was observed by Benjamin and Feir and, independently, by Whitham in Stokes' waves. I will discuss a general mechanism to study modulational instability of periodic traveling waves which can be applied to several classes of nonlinear dispersive equations including KdV, BBM, and regularized Boussinesq type equations.<br />
<br />
<br />
===Khai Nguyen===<br />
<br />
Title: Burgers Equation with Some Nonlocal Sources<br />
<br />
Abstract: Consider the Burgers equation with some nonlocal sources, which were derived from models of nonlinear wave with constant frequency. This talk will present some recent results on the global existence of entropy weak solutions, priori estimates, and a uniqueness result for both Burgers-Poisson and Burgers-Hilbert equations. Some open questions will be discussed.<br />
<br />
===Hongwei Gao=== <br />
<br />
Title: Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations<br />
<br />
Abstract: In this talk, we discuss the stochastic homogenization of certain nonconvex Hamilton-Jacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of (level-set) convex Hamiltonians and a sequence of (level-set) concave Hamiltonians through the min-max formula. We provide a monotonicity assumption on the contact values between those stably paired Hamiltonians so as to guarantee the stochastic homogenization. If time permits, we will talk about some homogenization results when the monotonicity assumption breaks down.<br />
<br />
===Huy Nguyen===<br />
<br />
Title : Compressible fluids and active potentials<br />
<br />
Abstract: We consider a class of one dimensional compressible systems with degenerate diffusion coefficients. We establish the fact that the solutions remain smooth as long as the diffusion coefficients do not vanish, and give local and global existence results. The models include the barotropic compressible Navier-Stokes equations, shallow water systems and the lubrication approximation of slender jets. In all these models the momentum equation is forced by the gradient of a solution-dependent potential: the active potential. The method of proof uses the Bresch-Desjardins entropy and the analysis of the evolution of the active potential.<br />
<br />
===In-Jee Jeong===<br />
<br />
Title: Singularity formation for the 3D axisymmetric Euler equations<br />
<br />
Abstract: We consider the 3D axisymmetric Euler equations on exterior domains $\{ (x,y,z) : (1 + \epsilon|z|)^2 \le x^2 + y^2 \} $ for any $\epsilon > 0$ so that we can get arbitrarily close to the exterior of a cylinder. We construct a strong local well-posedness class, and show that within this class there exist compactly supported initial data which blows up in finite time. The local well-posedness class consists of velocities which are uniformly Lipschitz in space and have finite energy. Our results were inspired by recent works of Hou-Luo, Kiselev-Sverak, and many others, and the proof builds up on our previous works on 2D Euler and Boussinesq systems. This is joint work with Tarek Elgindi.<br />
<br />
===Jeff Calder===<br />
<br />
Title: Nonlinear PDE continuum limits in data science and machine learning<br />
<br />
Abstract: We will present some recent results on PDE continuum limits for (random) discrete problems in data science and machine learning. All of the problems satisfy a type of discrete comparison/maximum principle and so the continuum PDEs are properly interpreted in the viscosity sense. We will present results for nondominated sorting, convex hull peeling, and graph-based semi-supervised learning. Nondominated sorting is an algorithm for arranging points in Euclidean space into layers by repeatedly peeling away coordinatewise minimal points, and the continuum PDE turns out to be a Hamilton-Jacobi equation. Convex hull peeling is used to order data by repeatedly peeling the vertices of the convex hull, and the continuum limit is motion by a power of Gauss curvature. Finally, a recently proposed class of graph-based learning problems have PDE continuum limits corresponding to weighted p-Laplace equations. In each case the continuum PDEs provide insights into the data science/engineering problems, and suggest avenues for fast approximate algorithms based on the PDE interpretations.<br />
<br />
===Hitoshi Ishii===<br />
<br />
Title: Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory<br />
<br />
Abstract: In the lecture, I discuss two asymptotic problems related to Hamilton-Jacobi equations. One concerns the long-time behavior of solutions of time evolutionary Hamilton-Jacobi equations and the other is the so-called vanishing discount problem for stationary Hamilton-Jacobi equations. The last two decades have seen a fundamental importance of weak KAM theory in the asymptotic analysis of Hamilton-Jacobi equations. I explain briefly the Aubry sets and Mather measures from weak KAM theory and their use in the analysis of the two asymptotic problems above.<br />
<br />
===Mihaela Ifrim===<br />
<br />
Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation<br />
<br />
Our goal is to take a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation. While this problem is known to be both completely integrable and globally well-posed in $L^2$, much less seems to be known concerning its long time dynamics. We present that for small localized data the solutions have (nearly) dispersive dynamics almost globally in time. An additional objective is to revisit the $L^2$ theory for the Benjamin-Ono equation and provide a simpler, self-contained approach. This is joined work with Daniel Tataru.<br />
<br />
===Longjie Zhang===<br />
<br />
On curvature flow with driving force starting as singular initial curve in the plane<br />
<br />
We consider a family of axisymmetric curves evolving by its mean curvature with driving force in the plane. However, the initial curve is oriented singularly at origin. We investigate this problem by level set method and give some criteria to judge whether the interface evolution is fattening or not. In the end, we can classify the solutions into three categories and provide the asymptotic behavior in each category. Our main tools in this paper are level set method and intersection number principle.<br />
<br />
===Jaeyoung Byeon===<br />
<br />
Title: Patterns formation for elliptic systems with large interaction forces<br />
<br />
Abstract: Nonlinear elliptic systems arising from nonlinear Schroedinger systems have simple looking reaction terms. The corresponding energy for the reaction terms can be expressed as quadratic forms in terms of density functions. The i, j-th entry of the matrix for the quadratic form represents the interaction force between the components i and j of the system. If the sign of an entry is positive, the force between the two components is attractive; on the other hand, if it is negative, it is repulsive. When the interaction forces between different components are large, the network structure of attraction and repulsion between components might produce several interesting patterns for solutions. As a starting point to study the general pattern formation structure for systems with a large number of components, I will first discuss the simple case of 2-component systems, and then the much more complex case of 3-component systems.<br />
<br />
<br />
===Tuoc Phan===<br />
Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application.<br />
<br />
Abstract: In this talk, we first introduce a problem on the existence of global time smooth solutions for a system of cross-diffusion equations. We then recall some classical results on regularity theories, and show that to solve our problem, new results on regularity theory estimates of Calderon-Zygmund type for gradients of solutions to a class of parabolic equations in Lebesgue spaces are required. We then discuss a result on Calderon-Zygmnud type estimate in the concrete setting to solve our<br />
mentioned problem regarding the system of cross-diffusion equations. The remaining part of the talk will be focused on some new generalized results on regularity gradient estimates for some general class of quasi-linear parabolic equations. Regularity estimates for gradients of solutions in Lorentz spaces will be presented. Ideas of the proofs for the results are given.<br />
<br />
===Hiroyoshi Mitake===<br />
Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can be formally derived by the singular limit of the weakly coupled system of the Allen-Cahn equation. By using the level set approach, this system can be written as a quasi-monotone degenerate parabolic system. We give results of the well-posedness of viscosity solutions, and study the singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
<br />
===Dongnam Ko===<br />
On the emergence of local flocking phenomena in Cucker-Smale ensembles<br />
<br />
Emergence of flocking groups are often observed in many complex network systems. The Cucker-Smale model is one of the flocking model, which describes the dynamics of attracting particles. This talk concerns time-asymptotic behaviors of Cucker-Smale particle ensembles, especially for mono-cluster and bi-cluster flockings. The emergence of flocking phenomena is determined by sufficient initial conditions, coupling strength, and communication weight decay. Our asymptotic analysis uses the Lyapunov functional approach and a Lagrangian formulation of the coupled system. We derive a system of differential inequalities for the functionals that measure the local fluctuations and group separations along particle trajectories. The bootstrapping argument is the key idea to prove the gathering and separating behaviors of Cucker-Smale particles simultaneously.<br />
<br />
===Sameer Iyer===<br />
Title: Global-in-x Steady Prandtl Expansion over a Moving Boundary.<br />
<br />
Abstract: I will outline the proof that steady, incompressible Navier-Stokes flows posed over the moving boundary, y = 0, can be decomposed into Euler and Prandtl flows globally in the tangential variable, assuming a sufficiently small velocity mismatch. The main obstacles in the analysis center around obtaining sharp decay rates for the linearized profiles and the remainders. The remainders are controlled via a high-order energy method, supplemented with appropriate embedding theorems, which I will present.<br />
<br />
===Jingrui Cheng===<br />
<br />
A 1-D semigeostrophic model with moist convection.<br />
<br />
We consider a simplified 1-D model of semigeostrophic system with moisture, which describes moist convection in a single column in the atmosphere. In general, the solution is non-continuous and it is nontrivial part of the problem to find a suitable definition of weak solutions. We propose a plausible definition of such weak solutions which describes the evolution of the probability distribution of the physical quantities, so that the equations hold in the sense of almost everywhere. Such solutions are constructed from a discrete scheme which obeys the physical principles. This is joint work with Mike Cullen, together with Bin Cheng, John Norbury and Matthew Turner.<br />
<br />
===Donghyun Lee===<br />
<br />
We construct a unique global-in-time solution to the Vlasov-Poisson-Boltzmann system in convex domains with the diffuse boundary condition. Moreover we prove an exponential convergence of distribution function toward the global Maxwellian.<br />
<br />
===Myoungjean Bae===<br />
<br />
3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system.<br />
<br />
I will present a recent result on the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler–Poisson system in a cylinder supplemented with non-small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl (=angular momentum density) component. This talk is based on a joint work with S. Weng (Wuhan University, China).<br />
<br />
===Jingchen Hu===<br />
<br />
Shock Reflection and Diffraction Problem with Potential Flow Equation<br />
<br />
In this talk, we will present our work on nonsymmetric shock reflection and diffraction problem, the equation concerned is potential flow equation, which is a simplification of Euler System, mainly based on the assumption that flow has no vortex. We showed in both nonsymmetric reflection case and diffraction case, that physically admissible solution does not exist. This implies that the formation of vortex is essential to maintain the structural stability of shock reflection and diffraction.<br />
<br />
===Xiaoqin Guo===<br />
<br />
Quantitative homogenization and Harnack inequality for a degenerate discrete nondivergence form random operator.<br />
<br />
In the d-dimensional integer lattice $\mathbb Z^d$, $d\ge 2$, we consider a discrete non-divergence form difference operator<br />
$$<br />
L_a u(x)=\sum_{i=1}^d a_i(x)[u(x+e_i)+u(x-e_i)-2u(x)]<br />
$$<br />
where $a(x)=diag(a_1(x),..., a_d(x)), x\in\mathbb Z^d$ are random nonnegative diagonal matrices which are identically distributed and independent and with a positive expectation.<br />
A difficulty in studying this problem is that coefficients are allowed to be zero. In this talk, using random walks in random media and its percolative structure, we will present a Harnack inequality and quantitative homogenization result for this random operator. Joint work with N.Berger, M.Cohen and J.-D. Deuschel.<br />
<br />
===Ru-Yu Lai===<br />
<br />
Inverse problems for Maxwell's equations and its application.<br />
<br />
This talk will illustrate the application of complex geometrical optics (CGO) solutions to Maxwell's equations. <br />
First, I will explain the increasing stability behavior of coefficients for Maxwell equations. <br />
In particular, by using CGO solutions, the stability estimate of the conductivity is improving when frequency is growing. <br />
Second, I will describe the construction of new families of accelerating and almost nondiffracting beams for Maxwell's equations. <br />
They have the form of wave packets that propagate along circular trajectories while almost preserving a trasverse intensity profile.<br />
<br />
===Norbert Pozar===<br />
<br />
<br />
Viscosity solutions for the crystalline mean curvature flow<br />
<br />
In this talk I will present some recent results concerning the analysis<br />
of the level set formulation of the crystalline mean curvature flow.<br />
The crystalline mean curvature, understood as the first variation of an<br />
anisotropic surface energy with an anisotropy whose Wulff shape is a<br />
polytope, is a singular quantity, nonlocal on the flat parts of the<br />
crystal surface. Therefore the level set equation is not a usual PDE and<br />
does not fit into the classical viscosity solution framework. Its<br />
well-posedness in dimensions higher than two was an open problem until<br />
very recently. In a joint work with Yoshikazu Giga (U. of Tokyo), we<br />
introduce a new notion of viscosity solutions for this problem and<br />
establish its well-posedness for compact crystals in an arbitrary<br />
dimension, including a comparison principle and the stability with<br />
respect to approximation by a smooth anisotropic mean curvature flow.<br />
<br />
<br />
<br />
===Sigurd Angenent===<br />
The Huisken-Hamilton-Gage theorem on compact convex solutions to MCF shows that in forward time all solutions do the same thing, namely, they shrink to a point and become round as they do so. Even though MCF is ill-posed in backward time there do exist solutions that are defined for all t<0 , and one can try to classify all such &ldquo;Ancient Solutions.&rdquo; In doing so one finds that there is interesting dynamics associated to ancient solutions. I will discuss what is currently known about these solutions. Some of the talk is based on joint work with Sesum and Daskalopoulos.<br />
<br />
===Serguei Denissov===<br />
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.<br />
<br />
<br />
===Bing Wang===<br />
We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R3. This is a joint work with H.Z. Li.<br />
<br />
===Eric Baer===<br />
We discuss a recent result showing that a characterization of isoperimetric sets (that is, sets minimizing a relative perimeter functional with respect to a fixed volume constraint) inside convex cones as sections of balls centered at the origin (originally due to P.L. Lions and F. Pacella) remains valid for a class of "almost-convex" cones. Key tools include compactness arguments and the use of classically known sharp characterizations of lower bounds for the first nonzero Neumann eigenvalue associated to (geodesically) convex domains in the hemisphere. The work we describe is joint with A. Figalli.<br />
<br />
===Ben Seeger===<br />
I present a homogenization result for pathwise Hamilton-Jacobi equations with "rough" multiplicative driving signals. In doing so, I derive a new well-posedness result when the Hamiltonian is smooth, convex, and positively homogenous. I also demonstrate that equations involving multiple driving signals may homogenize or exhibit blow-up.<br />
<br />
===Sona Akopian===<br />
Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.<br />
<br />
We solve the Cauchy problem associated to an epsilon-parameter family of homogeneous Boltzmann equations for very soft and Coulomb potentials. Proposed in 2013 by Bobylev and Potapenko, the collision kernel that we use is a Dirac mass concentrated at very small angles and relative speeds. The main advantage of such a kernel is that it does not separate its variables (relative speed $u$ and scattering angle $\theta$) and can be viewed as a pseudo-Maxwell molecule collision kernel, which allows for the splitting of the Boltzmann collision operator into its gain and loss terms. Global estimates on the gain term gives us an existence theory for $L^1_k \capL^p$ with any $k\geq 2$ and $p\geq 1.$ Furthermore the bounds we obtain are independent of the epsilon parameter, which allows for analysis of the solutions in the grazing collisions limit, i.e., when epsilon approaches zero and the Boltzmann equation becomes the Landau equation. <br />
<br />
===Sylvia Serfaty===<br />
Mean-Field Limits for Ginzburg-Landau vortices<br />
<br />
Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.<br />
<br />
<br />
===Gui-Qiang Chen===<br />
Supersonic Flow onto Solid Wedges, Multidimensional Shock Waves and Free Boundary Problems<br />
<br />
When an upstream steady uniform supersonic flow, governed by the Euler equations,<br />
impinges onto a symmetric straight-sided wedge, there are two possible steady oblique shock<br />
configurations if the wedge angle is less than the detachment angle -- the steady weak shock<br />
with supersonic or subsonic downstream flow (determined by the wedge angle that is less or larger<br />
than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which<br />
satisfy the entropy conditions.<br />
The fundamental issue -- whether one or both of the steady weak and strong shocks are physically<br />
admissible solutions -- has been vigorously debated over the past eight decades.<br />
In this talk, we discuss some of the most recent developments on the stability analysis<br />
of the steady shock solutions in both the steady and dynamic regimes.<br />
The corresponding stability problems can be formulated as free boundary problems<br />
for nonlinear partial differential equations of mixed elliptic-hyperbolic type, whose<br />
solutions are fundamental for multidimensional hyperbolic conservation laws.<br />
Some further developments, open problems, and mathematical challenges in this direction<br />
are also addressed.<br />
<br />
===Zhenfu Wang===<br />
<br />
Title: Mean field limit for stochastic particle systems with singular forces<br />
<br />
Abstract: We consider large systems of particles interacting through rough interaction kernels. We are able to control the relative entropy between the N-particles distribution<br />
and the expected limit which solves the corresponding McKean-Vlasov PDE. This implies the Mean Field limit to the McKean-Vlasov system together with Propagation of Chaos<br />
through the strong convergence of all the marginals. The method works at the level of the Liouville equation and relies on precise combinatorics results.<br />
<br />
===Andrei Tarfulea===<br />
We consider a model for three-dimensional fluid flow on the torus that also keeps track of the local temperature. The momentum equation is the same as for Navier-Stokes, however the kinematic viscosity grows as a function of the local temperature. The temperature is, in turn, fed by the local dissipation of kinetic energy. Intuitively, this leads to a mechanism whereby turbulent regions increase their local viscosity and<br />
dissipate faster. We prove a strong a priori bound (that would fall within the Ladyzhenskaya-Prodi-Serrin criterion for ordinary Navier-Stokes) on the thermally weighted enstrophy for classical solutions to the coupled system.<br />
<br />
===Siao-hao Guo===<br />
Analysis of Velázquez's solution to the mean curvature flow with a type II singularity<br />
<br />
Velázquez discovered a solution to the mean curvature flow which develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution, the rescaled flow converges in the C^0 sense to a minimal hypersurface which is tangent to Simons' cone at infinity. In this talk, we will present that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we will show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form. This is a joint work with N. Sesum.<br />
<br />
===Jianfeng Lu===<br />
Evolution of crystal surfaces: from mesoscopic to continuum models<br />
<br />
In this talk, we will discuss some of our recent results on understanding various models for crystal surface evolution at different physical scales; in particular, we will focus on the connection of mesoscopic and continuum (PDE) models for crystal surface relaxation and also discuss several PDEs arising from different physical scenarios. Many interesting open problems remain to be studied. Based on joint work with Yuan Gao, Jian-Guo Liu, Dio Margetis and Jeremy Marzuola.<br />
<br />
===Chris Henderson===<br />
A local-in-time Harnack inequality and applications to reaction-diffusion equations<br />
<br />
The classical Harnack inequality requires one to look back in time to obtain a uniform lower bound on the solution to a parabolic equation. In this talk, I will introduce a Harnack-type inequality that allows us to remove this restriction at the expense of a slightly weaker bound. I will then discuss applications of this bound to (time permitting) three non-local reaction-diffusion equations arising in biology. In particular, in each case, this inequality allows us to show that solutions to these equations, which do not enjoy a maximum principle, may be compared with solutions to a related local equation, which does enjoy a maximum principle. Precise estimates of the propagation speed follow from this.<br />
<br />
<br />
===Jeffrey Streets===<br />
Generalized Kahler Ricci flow and a generalized Calabi conjecture<br />
<br />
Generalized Kahler geometry is a natural extension of Kahler geometry with roots in mathematical physics, and is a particularly rich instance of Hitchin's program of `generalized geometries.' In this talk I will discuss an extension of Kahler-Ricci flow to this setting. I will formulate a natural Calabi-Yau type conjecture based on Hitchin/Gualtieri's definition of generalized Calabi-Yau equations, then introduce the flow as a tool for resolving this. The main result is a global existence and convergence result for the flow which yields a partial resolution of this conjecture, and which classifies generalized Kahler structures on hyperKahler backgrounds.<br />
<br />
<br />
===Daniel Spirn===<br />
<br />
Dipole Trajectories in Bose-Einstein Condensates<br />
<br />
Bose-Einstein condensates (BEC) are a state of matter in which supercooled atoms condense into the lowest possible quantum state. One interesting important feature of BECs are the presence of vortices that form when the condensate is stirred with lasers. I will discuss the behavior of these vortices, which interact with both the confinement potential and other vortices. I will also discuss a related inverse problem in which the features of the confinement can be extracted by the propagation of vortex dipoles.<br />
<br />
===Donghyun Lee===<br />
<br />
The Boltzmann equation with specular reflection boundary condition in convex domains<br />
<br />
I will present a recent work (https://arxiv.org/abs/1604.04342) with Chanwoo Kim on the global-wellposedness and stability of the Boltzmann equation in general smooth convex domains.<br />
<br />
===Kevin Zumbrun===<br />
<br />
TITLE: A Stable Manifold Theorem for a class of degenerate evolution equations <br />
<br />
ABSTRACT: We establish a Stable Manifold Theorem, with consequent exponential decay to equilibrium, for a class <br />
<br />
of degenerate evolution equations $Au'+u=D(u,u)$ with A bounded, self-adjoint, and one-to-one, but not invertible, and <br />
<br />
$D$ a bounded, symmetric bilinear map. This is related to a number of other scenarios investigated recently for which the <br />
<br />
associated linearized ODE $Au'+u=0$ is ill-posed with respect to the Cauchy problem. The particular case studied here <br />
<br />
pertains to the steady Boltzmann equation, yielding exponential decay of large-amplitude shock and boundary layers.<br />
<br />
<br />
<br />
===Will Feldman===<br />
<br />
Liquid Drops on a Rough Surface<br />
<br />
I will discuss the problem of determining the minimal energy shape of a liquid droplet resting on a rough solid surface. The shape of a liquid drop on a solid is strongly affected by the micro-structure of the surface on which it rests, where the surface inhomogeneity arises through varying chemical composition and surface roughness. I will explain a macroscopic regularity theory for the free boundary which allows to study homogenization, and more delicate properties like the size of the boundary layer induced by the surface roughness. <br />
<br />
The talk is based on joint work with Inwon Kim. A remark for those attending the weekend conference: this talk will attempt to have as little as possible overlap with I. Kim's conference talks. <br />
<br />
===Ryan Hynd===<br />
<br />
Extremal functions for Morrey’s inequality in convex domains<br />
<br />
A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor. We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold. <br />
<br />
===Gung-Min Gie ===<br />
<br />
Boundary layer analysis of some incompressible flows<br />
<br />
The motions of viscous and inviscid fluids are modeled respectively by the Navier-Stokes and Euler equations. Considering the Navier-Stokes equations at vanishing viscosity as a singular perturbation of the Euler equations, one major problem, still essentially open, is to verify if the Navier-Stokes solutions converge as the viscosity tends to zero to the Euler solution in the presence of physical boundary. In this talk, we study the inviscid limit and boundary layers of some simplified Naiver-Stokes equations by either imposing a certain symmetry to the flow or linearizing the model around a stationary Euler flow. For the examples, we systematically use the method of correctors proposed earlier by J. L. Lions and construct an asymptotic expansion as the sum of the Navier-Stokes solution and the corrector. The corrector, which corrects the discrepancies between the boundary values of the viscous and inviscid solutions, is in fact an (approximating) solution of the corresponding Prandtl type equations. The validity of our asymptotic expansions is then confirmed globally in the whole domain by energy estimates on the difference of the viscous solution and the proposed expansion. This is a joint work with J. Kelliher, M. Lopes Filho, A. Mazzucato, and H. Nussenzveig Lopes.<br />
<br />
===Tau Shean Lim===<br />
<br />
Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators<br />
<br />
We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) with ignition media f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the case of classical diffusion (i.e., Lu = Laplacian(u)) and non-local diffusion (Lu = J*u - u). Our work extends these results to general Levy operators. In particular, we show that a strong diffusivity in the underlying process (in the sense that the first moment of X_1 is infinite) prevents formation of fronts, while a weak diffusivity gives rise to a unique (up to translation) front U and speed c>0.<br />
<br />
===Tarek M. ELgindi===<br />
<br />
Propagation of Singularities in Incompressible Fluids<br />
<br />
We will discuss some recent results on the local and global stability of certain singular solutions to the incompressible 2d Euler equation. We will begin by giving a brief overview of the classical and modern results on the 2d Euler equation--particularly related to well-posedness theory in critical spaces. Then we will present a new well-posedness class which allows for merely Lipschitz continuous velocity fields and non-decaying vorticity. This will be based upon some interesting estimates for singular integrals on spaces with L^\infty scaling. After that we will introduce a class of scale invariant solutions to the 2d Euler equation and describe some of their remarkable properties including the existence of pendulum-like quasi periodic solutions and infinite-time cusp formation in vortex patches with corners. This is a joint work with I. Jeong. <br />
<br />
<br />
===Adrian Tudorascu===<br />
<br />
Hamilton-Jacobi equations in the Wasserstein space of probability measures <br />
<br />
In 2008 Gangbo, Nguyen and Tudorascu showed that certain variational solutions of the Euler-Poisson system in 1D can be regarded as optimal paths for the value-function giving the viscosity solution of some (infinite-dimensional) Hamilton-Jacobi equation whose phase-space is the Wasserstein space of Borel probability measures with finite second moment. At around the same time, Lasry, Lions, and others became interested in such Hamilton-Jacobi equations (HJE) in connection with their developing theory of Mean-Field games. A different approach (less intrinsic than ours) to the notion of viscosity solution was preferred, one that made an immediate connection between HJE in the Wasserstein space and HJE in Hilbert spaces (whose theory was well-studied and fairly well-understood). At the heart of the difference between these approaches lies the choice of the sub/supper-differential in the context of the Wasserstein space (i.e. the interpretation of ``cotangent space'' to this ``pseudo-Riemannian'' manifold) . In this talk I will start with a brief introduction to Mean-Field games and Optimal Transport, then I will discuss the challenges we encounter in the analysis of (our intrinsic) viscosity solutions of HJE in the Wasserstein space. Based on joint work with W. Gangbo.<br />
<br />
===Alexis Vasseur===<br />
<br />
Compressible Navier-Stokes equations with degenerate viscosities <br />
<br />
We will discuss recent results on the construction of weak solutions for<br />
3D compressible Navier-Stokes equations with degenerate viscosities.<br />
The method is based on the Bresch and Desjardins entropy. The main<br />
contribution is to derive MV type inequalities for the weak solutions,<br />
even if it is not verified by the first level of approximation. This<br />
provides existence of global solutions in time, for the compressible<br />
Navier-Stokes equations, in three dimensional space, with large initial<br />
data, possibly vanishing on the vacuum.<br />
<br />
===Minh-Binh Tran===<br />
<br />
Quantum kinetic problems<br />
<br />
After the production of the first BECs, there has been an explosion of research on the kinetic theory associated to BECs. Later, Gardinier, Zoller and collaborators derived a Master Quantum Kinetic Equation for BECs and introduced the terminology ”Quantum Kinetic Theory”. In 2012, Reichl and collaborators made a breakthrough in discovering a new collision operator, which had been missing in the previous works.<br />
My talk is devoted to the description of our recent mathematical works on quantum kinetic theory. The talk will be based on my joint works with Alonso, Gamba (existence, uniqueness, propagation of moments), Nguyen (Maxwellian lower bound), Soffer (coupling Schrodinger–kinetic equations), Escobedo (convergence to equilibrium), Craciun (the analog between the global attractor conjecture in chemical reaction network and the convergence to equilibrium of quantum kinetic equations), Reichl (derivation).<br />
<br />
===David Kaspar===<br />
<br />
Kinetics of shock clustering<br />
<br />
Suppose we solve a (deterministic) scalar conservation law<br />
with random initial data. Can we describe the probability law of the<br />
solution as a stochastic process in x for fixed later time t? The<br />
answer is yes, for certain Markov initial data, and the probability<br />
law factorizes as a product of kernels. These kernels are obtained by<br />
solving a mean-field kinetic equation which most closely resembles the<br />
Smoluchowski coagulation equation. We discuss prior and ongoing work<br />
concerning this and related problems.<br />
<br />
===Brian Weber===<br />
<br />
Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds<br />
<br />
Understanding scalar-flat instantons is crucial for knowing how Ka ̈hler manifolds degenerate. It is known that scalar-flat Kahler 4-manifolds with two symmetries give rise to a pair of linear degenerate-elliptic Heston type equations <br />
of the form x(fxx + fyy) + fx = 0, which were originally studied in mathematical finance. Vice- versa, solving these PDE produce scalar-flat Kahler 4-manifolds. These PDE have been studied locally, but here we describe new global results <br />
and their implications, partic- ularly a classification of scalar-flat metrics on K ̈ahler 4-manifolds and applications for the study of constant scalar curvature and extremal Ka ̈hler metrics.<br />
<br />
<br />
<br />
<br />
<br />
===Tianling Jin===<br />
<br />
Holder gradient estimates for parabolic homogeneous p-Laplacian equations<br />
<br />
We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation <br />
u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u),<br />
where 1<p<\infty. This equation arises from tug-of-war like stochastic games with white noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.<br />
<br />
===Russell Schwab===<br />
<br />
Neumann homogenization via integro-differential methods<br />
<br />
In this talk I will describe how one can use integro-differential methods to attack some Neumann homogenization problems-- that is, describing the effective behavior of solutions to equations with highly oscillatory Neumann data. I will focus on the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with non-co-normal oscillatory Neumann conditions. The analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain. This is joint work with Nestor Guillen.<br />
<br />
===Jingrui Cheng===<br />
<br />
Semi-geostrophic system with variable Coriolis parameter.<br />
<br />
The semi-geostrophic system (abbreviated as SG) is a model of large-scale atmospheric/ocean flows. Previous works about the SG system have been restricted to the case of constant Coriolis force, where we write the equation in "dual coordinates" and solve. This method does not apply for variable Coriolis parameter case. We develop a time-stepping procedure to overcome this difficulty and prove local existence and uniqueness of smooth solutions to SG system. This is joint work with Michael Cullen and Mikhail Feldman.<br />
<br />
<br />
===Paul Rabinowitz===<br />
<br />
On A Double Well Potential System<br />
<br />
We will discuss an elliptic system of partial differential equations of the form<br />
\[<br />
-\Delta u + V_u(x,u) = 0,\;\;x \in \Omega = \R \times \mathcal{D}\subset \R^n, \;\;\mathcal{D} \; bounded \subset \R^{n-1}<br />
\]<br />
\[<br />
\frac{\partial u}{\partial \nu} = 0 \;\;on \;\;\partial \Omega,<br />
\]<br />
with $u \in \R^m$,\; $\Omega$ a cylindrical domain in $\R^n$, and $\nu$ the outward pointing normal to $\partial \Omega$. <br />
Here $V$ is a double well potential with $V(x, a^{\pm})=0$ and $V(x,u)>0$ otherwise. When $n=1, \Omega =\R^m$ and \eqref{*} is a Hamiltonian system of ordinary differential equations. <br />
When $m=1$, it is a single PDE that arises as an Allen-Cahn model for phase transitions. We will <br />
discuss the existence of solutions of \eqref{*} that are heteroclinic from $a^{-}$ to $a^{+}$ or homoclinic to $a^{-}$,<br />
i.e. solutions that are of phase transition type.<br />
<br />
This is joint work with Jaeyoung Byeon (KAIST) and Piero Montecchiari (Ancona).<br />
<br />
===Hong Zhang===<br />
<br />
On an elliptic equation arising from composite material<br />
<br />
I will present some recent results on second-order divergence type equations with piecewise constant coefficients. This problem arises in the study of composite materials with closely spaced interface boundaries, and the classical elliptic regularity theory are not applicable. In the 2D case, we show that any weak solution is piecewise smooth without the restriction of the underling domain where the equation is satisfied. This completely answers a question raised by Li and Vogelius (2000) in the 2D case. Joint work with Hongjie Dong.<br />
<br />
===Aaron Yip===<br />
<br />
Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media<br />
<br />
The talk will describe some results on the behavior of solutions of motion by mean curvature in inhomogeneous media. Emphasis will be put on the pinning and de-pinning transition, continuum limit of discrete spin systems and the motion of interface between patterns.<br />
<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Selection problem for fully nonlinear equations<br />
<br />
Recently, there was substantial progress on the selection problem on the ergodic problem for Hamilton-Jacobi equations, which was open during almost 30 years. In the talk, I will first show a result on the convex Hamilton-Jacobi equation, then tell important problems which still remain. Next, I will mainly focus on a recent joint work with H. Ishii (Waseda U.), and H. V. Tran (U. Wisconsin-Madison) which is about the selection problem for fully nonlinear, degenerate elliptic partial differential equations. I will present a new variational approach for this problem.<br />
<br />
===Nestor Guillen===<br />
<br />
Min-max formulas for integro-differential equations and applications<br />
<br />
We show under minimal assumptions that a nonlinear operator satisfying what is known as a "global comparison principle" can be represented by a min-max formula in terms of very special linear operators (Levy operators, which involve drift-diffusion and integro-differential terms). Such type of formulas have been very useful in the theory of second order equations -for instance, by allowing the representation of solutions as value functions for differential games. Applications include results on the structure of Dirichlet-to-Neumann mappings for fully nonlinear second order elliptic equations.<br />
<br />
===Ryan Denlinger===<br />
<br />
The propagation of chaos for a rarefied gas of hard spheres in vacuum<br />
<br />
We are interested in the rigorous mathematical justification of<br />
Boltzmann's equation starting from the deterministic evolution of<br />
many-particle systems. O. E. Lanford was able to derive Boltzmann's<br />
equation for hard spheres, in the Boltzmann-Grad scaling, on a short<br />
time interval. Improvements to the time in Lanford's theorem have so far<br />
either relied on a small data hypothesis, or have been restricted to<br />
linear regimes. We revisit the small data regime, i.e. a sufficiently<br />
dilute gas of hard spheres dispersing into vacuum; this is a regime<br />
where strong bounds are available globally in time. Subject to the<br />
existence of such bounds, we give a rigorous proof for the propagation<br />
of Boltzmann's ``one-sided'' molecular chaos.<br />
<br />
===Misha Feldman===<br />
<br />
Shock reflection, free boundary problems and degenerate elliptic equations.<br />
<br />
Abstract: We will discuss shock reflection problem for compressible gas dynamics, and von Neumann<br />
conjectures on transition between regular and Mach reflections. We will discuss existence of solutions <br />
of regular reflection structure for potential flow equation, and also regularity of solutions, and<br />
properties of the shock curve (free boundary). Our approach is to reduce the shock reflection problem<br />
to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type. Open problems <br />
will also be discussed, including uniqueness.<br />
The talk is based on the joint works with Gui-Qiang Chen, Myoungjean Bae and Wei Xiang.<br />
<br />
===Jessica Lin===<br />
<br />
Optimal Quantitative Error Estimates in Stochastic<br />
Homogenization for Elliptic Equations in Nondivergence Form<br />
<br />
Abstract: I will present optimal quantitative error estimates in the<br />
stochastic homogenization for uniformly elliptic equations in<br />
nondivergence form. From the point of view of probability theory,<br />
stochastic homogenization is equivalent to identifying a quenched<br />
invariance principle for random walks in a balanced random<br />
environment. Under strong independence assumptions on the environment,<br />
the main argument relies on establishing an exponential version of the<br />
Efron-Stein inequality. As an artifact of the optimal error estimates,<br />
we obtain a regularity theory down to microscopic scale, which implies<br />
estimates on the local integrability of the invariant measure<br />
associated to the process. This talk is based on joint work with Scott<br />
Armstrong.<br />
<br />
===Sergey Bolotin===<br />
<br />
Degenerate billiards<br />
<br />
In an ordinary billiard trajectories of a Hamiltonian system<br />
are elastically reflected when colliding with a hypersurface (scatterer).<br />
If the scatterer is a submanifold of codimension more than one, then collisions are rare.<br />
Trajectories with infinite number of collisions form a lower dimensional dynamical system.<br />
Degenerate billiards appear as limits of ordinary billiards and in celestial mechanics.<br />
<br />
===Moon-Jin Kang===<br />
<br />
On contraction of large perturbation of shock waves, and inviscid limit problems<br />
<br />
This talk will start with the relative entropy method to handle the contraction of possibly large perturbations around viscous shock waves of conservation laws. In the case of viscous scalar conservation law in one space dimension, we obtain $L^2$-contraction for any large perturbations of shocks up to a Lipschitz shift depending on time. Such a time-dependent Lipschitz shift should be constructed from dynamics of the perturbation. In the case of multidimensional scalar conservation law, the perturbations of planar shocks are $L^2$-contractive up to a more complicated shift depending on both time and space variable, which solves a parabolic equation with inhomogeneous coefficient and force terms reflecting the perturbation. As a consequence, the $L^2$-contraction property implies the inviscid limit towards inviscid shock waves. At the end, we handle the contraction properties of admissible discontinuities of the hyperbolic system of conservation laws equipped with a strictly convex entropy.<br />
<br />
<br />
<br />
===Hung Tran===<br />
<br />
Some inverse problems in periodic homogenization of Hamilton--Jacobi equations.<br />
<br />
Abstract: We look at the effective Hamiltonian $\overline{H}$ associated with the Hamiltonian $H(p,x)=H(p)+V(x)$ in the periodic homogenization theory. Our central goal is to understand the relation between $V$ and $\overline{H}$. We formulate some inverse problems concerning this relation. Such type of inverse problems are in general very challenging. I will discuss some interesting cases in both convex and nonconvex settings. Joint work with Songting Luo and Yifeng Yu.<br />
<br />
<br />
===Eric Baer===<br />
<br />
Optimal function spaces for continuity of the Hessian determinant as a distribution.<br />
<br />
Abstract: In this talk we describe a new class of optimal continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on $\mathbb{R}^N$, obtained in collaboration with D. Jerison. Inspired by recent work of Brezis and Nguyen on the distributional Jacobian determinant, we show that the action is continuous on the Besov space $B(2-2/N,N)$ of fractional order, and that all continuity results in this scale of Besov spaces are consequences of this result. A key ingredient in the argument is the characterization of $B(2-2/N,N)$ as the space of traces of functions in the Sobolev space $W^{2,N}(\mathbb{R}^{N+2})$ on the subspace $\mathbb{R}^N$ (of codimension 2). The most elaborate part of the analysis is the construction of a counterexample to continuity in $B(2-2/N,p)$ with $p>N$. Tools involved in this step include the choice of suitable ``atoms" having a tensor product structure and Hessian determinant of uniform sign, formation of lacunary series of rescaled atoms, and delicate estimates of terms in the resulting multilinear expressions.<br />
<br />
===Donghyun Lee===<br />
<br />
FLUIDS WITH FREE-SURFACE AND VANISHING VISCOSITY LIMIT.<br />
<br />
Abstract : Free-boundary problems of incompressible fluids have been studied for several decades. In the viscous case, it is basically solved by Stokes regularity. However, the inviscid case problem is generally much harder, because the problem is purely hyperbolic. In this talk, we approach the problem via vanishing viscosity limit, which is a central problem of fluid mechanics. To correct boundary layer behavior, conormal Sobolev space will be introduced. In the spirit of the recent work by N.Masmoudi and F.Rousset (2012, non-surface tension), we will see how to get local regularity of incompressible free-boundary Euler, taking surface tension into account. This is joint work with Tarek Elgindi.<br />
If possible, we also talk about applying the similar technique to the free-boundary MHD(Magnetohydrodynamics). Especially, we will see that strong zero initial boundary condition is still valid for this coupled PDE. For the general boundary condition (for perfect conductor), however, the problem is still open.<br />
<br />
===Hyung-Ju Hwang===<br />
<br />
The Fokker-Planck equation in bounded domains<br />
<br />
abstract: In this talk, we consider the initial-boundary value problem for the Fokker-Planck equation in an interval or in a bounded domain with absorbing boundary conditions. We discuss a theory of well-posedness of classical solutions for the problem as well as the exponential decay in time, hypoellipticity away from the singular set, and the Holder continuity of the solutions up to the singular set. This is a joint work with J. Jang, J. Jung, and J. Velazquez.<br />
<br />
===Minh-Binh Tran===<br />
<br />
Nonlinear approximation theory for kinetic equations<br />
<br />
Abstract: Numerical resolution methods for the Boltzmann equation plays a very important role in the practical a theoretical study of the theory of rarefied gas. The main difficulty in the approximation of the Boltzmann equation is due to the multidimensional structure of the Boltzmann collision operator. The major problem with deterministic numerical methods using to solve Boltzmann equation is that we have to truncate the domain or to impose nonphysical conditions to keep the supports of the solutions in the velocity space uniformly compact. I<br />
n this talk, we will introduce our new way to make the connection between nonlinear approximation theory and kinetic theory. Our nonlinear wavelet approximation is nontruncated and based on an adaptive spectral method associated with a new wavelet filtering technique. The approximation is proved to converge and preserve many properties of the homogeneous Boltzmann equation. The nonlinear approximation solves the equation without having to impose non-physics conditions on the equation.<br />
<br />
===Bob Jensen===<br />
<br />
Crandall-Lions Viscosity Solutions of Uniformly Elliptic PDEs<br />
<br />
Abstract: I will discuss C-L viscosity solutions of uniformly elliptic partial differential equations for operators with only measurable spatial regularity. E.g., $L[u] = \sum a_{i\,j}(x)\,D_{i\,j}u(x)$ where $a_{i\,j}(x)$ is bounded, uniformly elliptic, and measurable in $x$. In general there isn't a meaningful extension of the C-L viscosity solution definition to operators with measurable spatial dependence. But under uniform ellipticity there is a natural extension. Though there isn't a general comparison principle in this context, we will see that the extended definition is robust and uniquely characterizes the ``right" solutions for such problems.<br />
<br />
===Luis Silvestre===<br />
<br />
A priori estimates for integral equations and the Boltzmann equation.<br />
<br />
Abstract: We will review some results on the regularity of general parabolic integro-differential equations. We will see how these results can be applied in order to obtain a priori estimates for the Boltzmann equation (without cutoff) modelling the evolution of particle density in a dilute gas. We derive a bound in L^infinity for the full Boltzmann equation, and Holder continuity estimates in the space homogeneous case.<br />
<br />
===Connor Mooney===<br />
<br />
Counterexamples to Sobolev regularity for degenerate Monge-Ampere equations<br />
<br />
Abstract: W^{2,1} estimates for the Monge-Ampere equation \det D^2u = f in R^n were first obtained by De Philippis and Figalli in the case that f is bounded between positive constants. Motivated by applications to the semigeostrophic equation, we consider the case that f is bounded but allowed to be zero on some set. In this case there are simple counterexamples to W^{2,1} regularity in dimension n \geq 3 that have a Lipschitz singularity. In contrast, if n = 2 a classical theorem of Alexandrov on the propagation of Lipschitz singularities shows that solutions are C^1. We will discuss a counterexample to W^{2,1} regularity in two dimensions whose second derivatives have nontrivial Cantor part, and also a related result on the propagation of Lipschitz / log(Lipschitz) singularities that is optimal by example.<br />
<br />
===Javier Gomez Serrano===<br />
<br />
Existence and regularity of rotating global solutions for active scalars<br />
<br />
A particular kind of weak solutions for a 2D active scalar equation are the so called patches, i.e., solutions for which the scalar is a step function taking one value inside a moving region and another in the complement. The evolution of such distribution is completely determined by the evolution of the boundary, allowing the problem to be treated as a non-local one dimensional equation for the contour. In this talk we will discuss the existence and regularity of uniformly rotating solutions for the vortex patch and generalized surface quasi-geostrophic (gSQG) patch equation. We will also outline the proof for the smooth (non patch) SQG case. Joint work with Angel Castro and Diego Cordoba.<br />
<br />
===Yifeng Yu===<br />
<br />
G-equation in the modeling of flame propagation.<br />
<br />
Abstract: G-equation is a well known model in turbulent combustion. In<br />
this talk, I will present joint works with Jack Xin about how the<br />
effective burning velocity (turbulent flame speed) depends on the strength<br />
of the ambient fluid (e.g. the speed of the wind) under various G-equation<br />
model.<br />
<br />
===Nam Le===<br />
<br />
Global smoothness of the Monge-Ampere eigenfunctions<br />
<br />
Abstract:<br />
In this talk, I will discuss global smoothness of the eigenfunctions of the Monge-Ampere operator on smooth, bounded and uniformly convex domains in all dimensions. A key ingredient in our analysis is boundary Schauder estimates for certain degenerate Monge-Ampere equations. This is joint work with Ovidiu Savin.<br />
<br />
===Qin Li===<br />
<br />
Kinetic-fluid coupling: transition from the Boltzmann to the Euler<br />
<br />
Abstract: Kinetic equations (the Boltzmann, the neutron transport equation etc.) are known to converge to fluid equations (the Euler, the heat equation etc.) in certain regimes, but when kinetic and fluid regime co-exist, how to couple the two systems remains an open problem. The key is to understand the half-space problem that resembles the boundary layer at the interface. In this talk, I will present a unified proof for the well-posedness of a class of half-space equations with general incoming data, propose an efficient spectral solver, and utilize it to couple fluid with kinetics. Moreover, I will present complete error analysis for the proposed spectral solver. Numerical results will be shown to demonstrate the accuracy of the algorithm.<br />
<br />
===Lu Wang===<br />
<br />
Asymptotic Geometry of Self-shrinkers<br />
<br />
Abstract: In this talk, we will discuss some recent progress towards the conjectural asymptotic behaviors of two-dimensional self-shrinkers of mean curvature flow.<br />
<br />
===Christophe Lacave===<br />
<br />
Well-posedness for 2D Euler in non-smooth domains<br />
<br />
The well-posedness of the Euler system has been of course the matter of many works, but a common point in all the previous studies is that the boundary is at least $C^{1,1}$. In a first part, we will establish the existence of global weak solutions of the 2D incompressible Euler equations for a large class of non-smooth open sets. Existence of weak solutions with $L^p$ vorticity is deduced from an approximation argument, that relates to the so-called $\gamma$-convergence of domains. In a second part, we will prove the uniqueness if the open set is the interior or the exterior of a simply connected domain, where the boundary has a finite number of corners. Although the velocity blows up near these corners, we will get a similar theorem to the Yudovich's result. Theses works are in collaboration with David Gerard-Varet, Evelyne Miot and Chao Wang.<br />
<br />
<br />
===Jun Kitagawa (Toronto)===<br />
<br />
Regularity theory for generated Jacobian equations: from optimal transport to geometric optics<br />
<br />
Equations of Monge-Ampere type arise in numerous contexts, and solutions often exhibit very subtle qualitative and quantitative properties; this is owing to the highly nonlinear nature of the equation, and its degeneracy (in the sense of ellipticity). Motivated by an example from geometric optics, I will talk about the class of Generated Jacobian Equations; recently introduced by Trudinger, this class also encompasses, for example, optimal transport, the Minkowski problem, and the classical Monge-Ampere equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field reflector problems (of interest from a physical and practical point of view). This talk is based on joint works with N. Guillen.<br />
<br />
===Jessica Lin (Madison)===<br />
<br />
Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations<br />
<br />
We establish error estimates for the stochastic homogenization of fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media. Based on the approach of Armstrong and Smart in the elliptic setting, we construct a quantity which captures the geometric behavior of solutions to parabolic equations. The error estimates are shown to be of algebraic order. This talk is based on joint work with Charles Smart.<br />
<br />
<br />
===Yaguang Wang (Shanghai Jiao Tong)===<br />
<br />
Stability of Three-dimensional Prandtl Boundary Layers<br />
<br />
In this talk, we shall study the stability of the Prandtl boundary layer<br />
equations in three space variables. First, we obtain a well-posedness<br />
result of the three-dimensional Prandtl equations under some constraint on<br />
its flow structure. It reveals that the classical Burgers equation plays an<br />
important role in determining this type of flow with special structure,<br />
that avoids the appearance of the complicated secondary flow in the<br />
three-dimensional Prandtl boundary layers. Second, we give an instability<br />
criterion for the Prandtl equations in three space variables. Both of<br />
linear and nonlinear stability are considered. This criterion shows that<br />
the monotonic shear flow is linearly stable for the three dimensional<br />
Prandtl equations if and only if the tangential velocity field direction is<br />
invariant with respect to the normal variable, which is an exact complement<br />
to the above well-posedness result for a special flow. This is a joint work<br />
with Chengjie Liu and Tong Yang.<br />
<br />
<br />
===Benoit Pausader (Princeton)===<br />
<br />
Global smooth solutions for the Euler-Maxwell problem for electrons in 2 dimensions<br />
<br />
It is well known that pure compressible fluids tend to develop shocks, even from small perturbation. We study how self consistent electromagnetic fields can stabilize these fluids. In a joint work with A. Ionescu and Y. Deng, we consider a compressible fluid of electrons in 2D, subject to its own electromagnetic field and to a field created by a uniform background of positively charged ions. We show that small smooth and irrotational perturbations of a uniform background at rest lead to solutions that remain globally smooth, in contrast with neutral fluids. This amounts to proving small data global existence for a system of quasilinear Klein-Gordon equations with different speeds.<br />
<br />
<br />
===Haozhao Li (University of Science and Technology of China)===<br />
<br />
Regularity scales and convergence of the Calabi flow<br />
<br />
We define regularity scales to study the behavior of the Calabi flow. <br />
Based on estimates of the regularity scales, we obtain convergence theorems<br />
of the Calabi flow on extremal K\"ahler surfaces, under the assumption of global existence<br />
of the Calabi flow solutions. Our results partially confirm Donaldson’s conjectural picture for<br />
the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra<br />
assumption that the scalar curvature is uniformly bounded.<br />
<br />
<br />
===Jennifer Beichman (UW-Madison)===<br />
<br />
Nonstandard dispersive estimates and linearized water waves<br />
<br />
In this talk, we focus on understanding the relationship between the decay of a solution to the linearized water wave problem and its initial data. We obtain decay bounds for a class of 1D dispersive equations that includes the linearized water wave. These decay bounds display a surprising growth factor, which we show is sharp. A further exploration leads to a result relating singularities of the initial data at the origin in Fourier frequency to the regularity of the solution.<br />
<br />
===Ben Fehrman (University of Chicago)===<br />
<br />
On The Existence of an Invariant Measure for Isotropic Diffusions in Random Environments<br />
<br />
I will discuss the existence of a unique mutually absolutely continuous invariant measure for isotropic diffusions in random environment, of dimension at least three, which are small perturbations of Brownian motion satisfying a finite range dependence. This framework was first considered in the continuous setting by Sznitman and Zeitouni and in the discrete setting by Bricmont and Kupiainen. The results of this talk should be seen as an extension of their work.<br />
<br />
I will furthermore mention applications of this analysis to the stochastic homogenization of the related elliptic and parabolic equations with random oscillatory boundary data and, explain how the existence of an invariant measure can be used to prove a Liouville property for the environment. In the latter case, the methods were motivated by work in the discrete setting by Benjamini, Duminil-Copin, Kozma and Yadin.<br />
<br />
===Vera Hur=== <br />
<br />
Instabilities in nonlinear dispersive waves<br />
<br />
I will speak on the wave breaking and the modulational instability of nonlinear wave trains in dispersive media. I will begin by a gradient blowup proof for the Boussinesq-Whitham equations for water waves. I will then describe a variational approach to determine instability to long wavelength perturbations for a general class of Hamiltonian systems, allowing for nonlocal dispersion. I will discuss KdV type equations with fractional dispersion in depth. Lastly, I will explain an asymptotics approach for Whitham's equation for water waves, qualitatively reproducing the Benjamin-Feir instability of Stokes waves.<br />
<br />
===Sung-Jin Oh=== <br />
<br />
Global well-posedness of the energy critical Maxwell-Klein-Gordon equation<br />
<br />
The massless Maxwell-Klein-Gordon system describes the interaction between an electromagnetic field (Maxwell) and a charged massless scalar field (massless Klein-Gordon, or wave). <br />
In this talk, I will present a recent proof, joint with D. Tataru, of global well-posedness and scattering of this system for arbitrary finite energy data in the (4+1)-dimensional Minkowski space, in which the PDE is energy critical.<br />
<br />
===Yuan Lou=== <br />
<br />
Asymptotic behavior of the smallest eigenvalue of an elliptic operator and its applications to evolution of dispersal<br />
<br />
We investigate the effects of diffusion and drift on the smallest eigenvalue of an elliptic operator with zero Neumann boundary condition. Various asymptotic behaviors of the smallest eigenvalue, as diffusion and drift rates approach zero or infinity, are derived. As an application, these qualitative results yield some insight into the evolution of dispersal in heterogeneous environments.<br />
<br />
===Diego Cordoba===<br />
<br />
Global existence solutions and geometric properties of the SQG sharp front<br />
<br />
A particular kind of weak solutions for a 2D active scalar are the so called sharp fronts, i.e., solutions for which the scalar<br />
is a step function. The evolution of such distribution is completely determined by the evolution of the boundary, allowing the problem <br />
to be treated as a non-local one dimensional equation for the contour. In this setting we will present several analytical results for the surface <br />
quasi-geostrophic equation (SQG): the existence of convex $C^{\infinity}$ global rotating solutions, elliptical shapes are not rotating solutions (as opposed to 2D Euler equations) and the existence of convex solutions that lose their convexity in finite time.<br />
<br />
<br />
<br />
===Greg Kuperberg===<br />
''Cartan-Hadamard and the Little Prince.''<br />
<br />
===Steve Hofmann===<br />
''Quantitative Rectifiability and Elliptic Equations''<br />
<br />
A classical theorem of F. and M. Riesz states that for a simply connected domain in the complex plane with a rectifiable boundary, harmonic measure and arc length measure on the boundary are mutually absolutely continuous. On the other hand, an example of C. Bishop and P. Jones shows that the latter conclusion may fail, in the absence of some sort of connectivity hypothesis. In this talk, we discuss recent developments in an ongoing program to find scale-invariant, higher dimensional versions of the F. and M. Riesz Theorem, as well as converses. In particular, we discuss substitute results that continue to hold in the absence of any connectivity hypothesis.<br />
<br />
===Xiangwen Zhang===<br />
''Alexandrov's Uniqueness Theorem for Convex Surfaces''<br />
<br />
A classical uniqueness theorem of Alexandrov says that: a closed strictly convex twice differentiable surface in R3 is uniquely determined to within a parallel translation when one gives a proper function of the principle curvatures. We will talk about a PDE proof for this thorem, by using the maximal principle and weak uniqueness continuation theorem of Bers-Nirenberg. Moreover, a stability result related to the uniqueness problem will be mentioned. This is a joint work with P. Guan and Z. Wang.<br />
If time permits, we will also briefly introduce the idea of our recent work on Alexandrov’s theorems for codimension two submanifolds in spacetimes.<br />
<br />
===Xuwen Chen===<br />
''The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-body Evolution''<br />
<br />
We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is focusing and matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D N-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to infinity. We prove that the limiting structure of the density matrices counterbalances this diverging coefficient. We establish the convergence of the BBGKY sequence and hence the propagation of chaos for the focusing quantum many-body system. We derive rigorously the 1D focusing cubic NLS as the mean-field limit of this 3D focusing quantum many-body dynamic and obtain the exact 3D to 1D coupling constant.<br />
<br />
===Kyudong Choi===<br />
''Finite time blow up for 1D models for the 3D Axisymmetric Euler Equations the<br />
2D Boussinesq system''<br />
<br />
In connection with the recent proposal for possible singularity formation at the boundary for solutions of the 3d axi-symmetric incompressible Euler's equations / the 2D Boussinesq system (Luo and Hou, 2013), we study models for the dynamics at the boundary and show that they exhibit a finite-time blow-up from smooth data. This is joint work with T. Hou, A. Kiselev, G. Luo, V. Sverak, and Y. Yao.<br />
<br />
===Myoungjean Bae===<br />
''Recent progress on study of Euler-Poisson system''<br />
<br />
In this talk, I will present recent progress on the following subjects:<br />
(1) Smooth transonic flow of Euler-Poisson system;<br />
(2) Transonic shock of Euler-Poisson system.<br />
This talk is based on collaboration with Ben Duan, Chujing Xie and Jingjing Xiao<br />
<br />
===Philip Isett===<br />
"Hölder Continuous Euler Flows"<br />
<br />
Motivated by the theory of hydrodynamic turbulence, L. Onsager conjectured<br />
in 1949 that solutions to the incompressible Euler equations with Holder<br />
regularity less than 1/3 may fail to conserve energy. C. De Lellis and L.<br />
Székelyhidi, Jr. have pioneered an approach to constructing such irregular<br />
flows based on an iteration scheme known as convex integration. This<br />
approach involves correcting “approximate solutions" by adding rapid<br />
oscillations which are designed to reduce the error term in solving the<br />
equation. In this talk, I will discuss an improved convex integration<br />
framework, which yields solutions with Holder regularity 1/5-, as well as<br />
other related results.<br />
<br />
===Lei Wu===<br />
''Geometric Correction for Diffusive Expansion in Neutron Transport Equation''<br />
<br />
We revisit the diffusive limit of a steady neutron transport equation in a 2-D unit disk with one-speed velocity. The traditional method is Hilbert expansions and boundary layer analysis. We will carefully study the classical theory of the construction of boundary layers, and discuss the necessity and specific method to add the geometric correction.<br />
<br />
===Xuan Hien Nguyen===<br />
In the 1990's, Kapouleas and Traizet constructed new examples of minimal surfaces by desingularizing the intersection of existing ones, such as catenoids and planes, with Scherk surfaces. Using the same strategy, one can prove the existence of new self-translating and self-shrinking surfaces under mean curvature flow. In this talk, we will survey the results obtained so far and propose some generalization and simplification of the techniques.<br />
<br />
<br />
<br />
===Greg Drugan (U. of Washington)===<br />
''Construction of immersed self-shrinkers''<br />
<br />
Abstract: We describe a procedure for constructing immersed<br />
self-shrinking solutions to mean curvature flow. <br />
The self-shrinkers we construct have a rotational symmetry, and<br />
the construction involves a detailed study of geodesics in the<br />
upper-half plane with a conformal metric.<br />
This is a joint work with Stephen Kleene.<br />
<br />
===Guo Luo (Caltech)===<br />
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''<br />
<br />
Abstract:<br />
Whether the 3D incompressible Euler equations can develop a singularity in <br />
finite time from smooth initial data is one of the most challenging problems in <br />
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this <br />
long-standing open question from a numerical point of view, by presenting a class of <br />
potentially singular solutions to the Euler equations computed in axisymmetric <br />
geometries. The solutions satisfy a periodic boundary condition along the axial direction <br />
and no-flow boundary condition on the solid wall. The equations are discretized in space <br />
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially <br />
designed adaptive (moving) meshes that are dynamically adjusted to the evolving <br />
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the <br />
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and <br />
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a <br />
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and <br />
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity <br />
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup <br />
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and <br />
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also <br />
suggests that the blowing-up solution develops a self-similar structure near the point of <br />
the singularity, as the singularity time is approached.<br />
<br />
===Xiaojie Wang(Stony Brook)===<br />
''Uniqueness of Ricci flow solutions on noncompact manifolds''<br />
<br />
Abstract:<br />
Ricci flow is an important evolution equation of Riemannian metrics.<br />
Since it was introduced by R. Hamilton in 1982, it has greatly changed the landscape of riemannian geometry. One of the fundamental question about ricci flow is when is its solution to initial value problem unique. On compact manifold, with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when further restrictions are imposed to the solution. In this talk, we will discuss various conditions that guarantee the uniqueness. In particular, we will discuss in details with the following uniqueness result. Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, on $M\times [0,\epsilon]$ for some $\epsilon>0$, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.<br />
<br />
===Roman Shterenberg(UAB)===<br />
''Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems''<br />
<br />
Abstract: We present a review of the results in multidimensional periodic<br />
and almost-periodic spectral problems. We discuss some recent progress and<br />
old/new ideas used in the constructions. The talk is mostly based on the<br />
joint works with Yu. Karpeshina and L. Parnovski.<br />
<br />
===Antonio Ache(Princeton)===<br />
''Ricci Curvature and the manifold learning problem''<br />
<br />
Abstract: In the first half of this talk we will review several notions of coarse or weak<br />
Ricci Curvature on metric measure spaces which include the works of Lott-Villani, Sturm<br />
and Ollivier. The discussion of the notion of coarse Ricci curvature will serve as <br />
motivation for developing a method to estimate the Ricci curvature of a an embedded<br />
submaifold of Euclidean space from a point cloud which has applications to the Manifold<br />
Learning Problem. Our method is based on combining the notion of ``Carre du Champ"<br />
introduced by Bakry-Emery with a result of Belkin and Niyogi which shows that it is<br />
possible to recover the rough laplacian of embedded submanifolds of the Euclidean space<br />
from point clouds. This is joint work with Micah Warren.<br />
<br />
===Jean-Michel Roquejoffre (Toulouse)===<br />
''Front propagation in the presence of integral diffusion''<br />
<br />
Abstract: In many reaction-diffusion equations, where diffusion is<br />
given by a second order elliptic operator, the solutions<br />
will exhibit spatial transitions whose velocity is asymptotically<br />
linear in time. The situation can be different when the diffusion is of the<br />
integral type, the most basic example being the fractional Laplacian:<br />
the velocity can be time-exponential. We will explain why, and<br />
discuss several situations where this type of fast propagation<br />
occurs.<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Free Boundary Problem related to Euler-Poisson system''<br />
<br />
One dimensional analysis of Euler-Poisson system shows that when incoming <br />
supersonic flow is fixed, transonic shock can be represented as a monotone <br />
function of exit pressure. From this observation, we expect well-posedness <br />
of transonic shock problem for Euler-Poisson system when exit pressure is <br />
prescribed in a proper range. In this talk, I will present recent progress <br />
on transonic shock problem for Euler-Poisson system, which is formulated <br />
as a free boundary problem with mixed type PDE system. <br />
This talk is based on collaboration with Ben Duan(POSTECH), Chujing Xie(SJTU) <br />
and Jingjing Xiao(CUHK).<br />
<br />
===Changhui Tan (University of Maryland)===<br />
''Global classical solution and long time behavior of macroscopic flocking models''<br />
<br />
Abstract: Self-organized behaviors are very common in nature and human societies.<br />
One widely discussed example is the flocking phenomenon which describes<br />
animal groups emerging towards the same direction. Several models such<br />
as Cucker-Smale and Motsch-Tadmor are very successful in characterizing<br />
flocking behaviors. In this talk, we will discuss macroscopic representation<br />
of flocking models. These systems can be interpreted as compressible Eulerian <br />
dynamics with nonlocal alignment forcing. We show global existence of classical solutions and long time <br />
flocking behavior of the system, when initial profile satisfies a threshold condition. On the other hand, another set<br />
of initial conditions will lead to a finite time break down of the system. This<br />
is a joint work with Eitan Tadmor.<br />
<br />
===Hongjie Dong (Brown University)===<br />
''Parabolic equations in time-varying domains''<br />
<br />
Abstract: I will present a recent result on the Dirichlet boundary value<br />
problem for parabolic equations in time-varying domains. The equations are<br />
in either divergence or non-divergence form with boundary blowup low-order<br />
coefficients. The domains satisfy an exterior measure condition.<br />
<br />
===Hao Jia (University of Chicago)===<br />
''Long time dynamics of energy critical defocusing wave equation with<br />
radial potential in 3+1 dimensions.''<br />
<br />
Abstract: We consider the long term dynamics of radial solution to the<br />
above mentioned equation. For general potential, the equation can have a<br />
unique positive ground state and a number of excited states. One can expect<br />
that some solutions might stay for very long time near excited states<br />
before settling down to an excited state of lower energy or the ground<br />
state. Thus the detailed dynamics can be extremely complicated. However<br />
using the ``channel of energy" inequality discovered by T.Duyckaerts,<br />
C.Kenig and F.Merle, we can show for generic potential, any radial solution<br />
is asymptotically the sum of a free radiation and a steady state as time<br />
goes to infinity. This provides another example of the power of ``channel<br />
of energy" inequality and the method of profile decompositions. I will<br />
explain the basic tools in some detail. Joint work with Baoping Liu and<br />
Guixiang Xu.<br />
<br />
===Alexander Pushnitski (King's College)===<br />
''An inverse spectral problem for Hankel operators''<br />
<br />
Abstract:<br />
I will discuss an inverse spectral problem for a certain class of Hankel<br />
operators. The problem appeared in the recent work by P.Gerard and S.Grellier as a<br />
step towards description of evolution in a model integrable non-dispersive<br />
equation. Several features of this inverse problem make it strikingly (and somewhat<br />
mysteriously) similar to an inverse problem for Sturm-Liouville operators. I will<br />
describe the available results for Hankel operators, emphasizing this similarity.<br />
This is joint work with Patrick Gerard (Orsay).<br />
<br />
===Ronghua Pan (Georgia Tech)===<br />
''Compressible Navier-Stokes-Fourier system with temperature dependent dissipation''<br />
<br />
Abstract: From its physical origin such as Chapman-Enskog or Sutherland, the viscosity and<br />
heat conductivity coefficients in compressible fluids depend on absolute temperature<br />
through power laws. The mathematical theory on the well-posedness and regularity on this<br />
setting is widely open. I will report some recent progress on this direction,<br />
with emphasis on the lower bound of temperature, and global existence of<br />
solutions in one or multiple dimensions. The relation between thermodynamics laws<br />
and Navier-Stokes-Fourier system will also be discussed. This talk is based on joint works<br />
with Junxiong Jia and Weizhe Zhang.<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Transonic shocks for Euler-Poisson system and related problems''<br />
<br />
Abstract: Euler-Poisson system models various physical phenomena<br />
including the propagation of electrons in submicron semiconductor<br />
devices and plasmas, and the biological transport of ions for channel<br />
proteins. I will explain difference between Euler system and<br />
Euler-Poisson system and mathematical difficulties arising due to this<br />
difference. And, recent results about subsonic flow and transonic flow<br />
for Euler-Poisson system will be presented. This talk is based on<br />
collaboration with Ben Duan and Chunjing Xie.<br />
<br />
<br />
===Mike Cullen (Met. Office, UK)===<br />
''Modelling the uncertainty in predicting large-scale atmospheric circulations''<br />
<br />
Abstract: This talk describes work on quantifying the uncertainty in climate<br />
models by using data assimilation techniques; and in estimating the properties<br />
of large-scale atmospheric flows by using asymptotic limit solutions.<br />
<br />
===Mohammad Ghomi(Math. Georgia Tech)===<br />
<br />
Abstract: We show that every smooth closed curve C immersed in Euclidean<br />
3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers<br />
P of pairs of parallel tangent lines, I of inflections (or points of<br />
vanishing curvature), and V of vertices (or points of vanishing torsion) of<br />
C. The proof, which employs curve shortening flow with surgery, is based on<br />
corresponding inequalities for the numbers of double points, singularities,<br />
and inflections of closed curves in the real projective plane and the sphere<br />
which intersect every closed geodesic. These findings extend some classical<br />
results in curve theory including works of Moebius, Fenchel, and Segre, which<br />
is also known as Arnold's ``tennis ball theorem".<br />
<br />
===Wei Xiang (Oxford)===<br />
<br />
Abstract: The vertical shock which initially separates two<br />
piecewise constant Riemann data, passes the wedge from left to right,<br />
then shock diffraction phenomena will occur and the incident shock<br />
becomes a transonic shock. Here we study this problem on nonlinear<br />
wave system as well as on potential flow equations. The existence and<br />
the optimal regularity across sonic circle of the solutions to this<br />
problem is established. The comparison of these two systems is<br />
discussed, and some related open problems are proposed.<br />
<br />
===Adrian Tudorascu (West Virginia University)===<br />
<br />
Abstract: <br />
This chalk & board presentation will start with an interesting motivation for<br />
these systems (the simplest of which was shown to model adhesion<br />
dynamics/ballistic aggregation by Zeldovich in 1970 in a paper published in<br />
"Astronomy & Astrophysics"). A general existence result for one-dimensional<br />
pressureless Euler/Euler-Poisson systems with/without viscosity will be<br />
discussed. This is achieved via entropy solutions for some appropriate scalar<br />
conservation laws; I will show that these solutions encode all the information<br />
necessary to obtain solutions for the pressureless systems. I will also discuss<br />
the stability and uniqueness of solutions, which is obtained via a contraction<br />
principle in the Wasserstein metric. This is based on joint work with T. Nguyen<br />
(U. Akron).<br />
<br />
===Diego Cordoba (Madrid)===<br />
<br />
Abstract: We consider the evolution of an interface generated between two immiscible, <br />
incompressible and irrotational fluids. Specifically we study the Muskat equation (the <br />
interface between oil and water in sand) and water wave equation (interface between water <br />
and vacuum). For both equations we will study well-posedness and the existence of smooth <br />
initial data for which the smoothness of the interface breaks down in finite time. We <br />
will also discuss some open problems.<br />
<br />
===Fausto Ferrari (Bologna)===<br />
''Semilinear PDEs and some symmetry properties of stable solutions''<br />
<br />
I will deal with stable solutions of semilinear elliptic PDE's and some of their<br />
symmetry's properties. Moreover, I will introduce some weighted Poincaré<br />
inequalities obtained by combining the notion of stable solution with the<br />
definition of weak solution.<br />
<br />
===Arshak Petrosyan (Purdue)===<br />
''Nonuniqueness in a free boundary problem from combustion''<br />
<br />
We consider a parabolic free boundary problem with a fixed gradient condition<br />
which serves as a simplified model for the propagation of premixed equidiffusional<br />
flames. We give a rigorous justification of an example due to J.L. V ́azquez that <br />
the initial data in the form of two circular humps leads to the nonuniqueness of limit <br />
solutions if the supports of the humps touch at the time of their maximal expansion.<br />
<br />
This is a joint work with Aaron Yip.<br />
<br />
<br />
===Changyou Wang (U. of Kentucky)===<br />
''Phase transition for higher dimensional wells''<br />
<br />
For a potential function <math>F</math> that has two global minimum sets consisting of two compact connected<br />
Riemannian submanifolds in <math style="vertical-align=100%" >\mathbb{R}^k</math>, we consider the singular perturbation problem:<br />
<br />
Minimizing <math>\int \left(|\nabla u|^2+\frac{1}{\epsilon^2} F(u)\right)</math> under given Dirichlet boundary data.<br />
<br />
I will discuss a recent joint work with F.H.Lin and X.B.Pan on the asymptotic, as the parameter <math>\epsilon</math><br />
tends to zero, in terms of the area of minimal hypersurface interfaces, the minimal connecting energy, and<br />
the energy of minimizing harmonic maps into the phase manifolds under both Dirichlet and partially free boundary<br />
data. Our results in particular addressed the static case of the so-called Keller-Rubinstein-Sternberg problem.<br />
<br />
===Philippe LeFloch (Paris VI)===<br />
''Kinetic relations for undercompressive shock waves and propagating phase boundaries''<br />
<br />
I will discuss the existence and properties of shock wave solutions to nonlinear hyperbolic systems that are small-scale dependent and, especially, contain undercompressive shock waves or propagating phase boundaries. Regularization-sensitive patterns often arise in continuum physics, especially in complex fluid flows. The so-called kinetic relation is introduced to characterize the correct dynamics of these nonclassical waves, and is tied to a higher-order regularization induced by a more complete model that takes into account additional small-scale physics. In the present lecture, I will especially explain the techniques of Riemann problems, Glimm-type scheme, and total variation functionals adapted to nonclassical shock waves. <br />
<br />
<br />
<br />
===Irina Mitrea===<br />
''Boundary Value Problems for Higher Order Differential Operators''<br />
<br />
As is well known, many phenomena in engineering and mathematical physics<br />
can be modeled by means of boundary value problems for a certain elliptic<br />
differential operator L in a domain D.<br />
<br />
When L is a differential operator of second order a variety of tools<br />
are available for dealing with such problems including boundary integral<br />
methods,<br />
variational methods, harmonic measure techniques, and methods based on<br />
classical<br />
harmonic analysis. The situation when the differential operator has higher order<br />
(as is the case for instance with anisotropic plate bending when one<br />
deals with<br />
fourth order) stands in sharp contrast with this as only fewer options<br />
could be<br />
successfully implemented. Alberto Calderon, one of the founders of the<br />
modern theory<br />
of Singular Integral Operators, has advocated in the seventies the use<br />
of layer potentials<br />
for the treatment of higher order elliptic boundary value problems.<br />
While the<br />
layer potential method has proved to be tremendously successful in the<br />
treatment<br />
of second order problems, this approach is insufficiently developed to deal<br />
with the intricacies of the theory of higher order operators. In fact,<br />
it is largely<br />
absent from the literature dealing with such problems.<br />
<br />
In this talk I will discuss recent progress in developing a multiple<br />
layer potential<br />
approach for the treatment of boundary value problems associated with<br />
higher order elliptic differential operators. This is done in a very<br />
general class<br />
of domains which is in the nature of best possible from the point of<br />
view of<br />
geometric measure theory. <br />
<br />
<br />
===Panagiota Daskalopoulos (Columbia U)===<br />
''Ancient solutions to geometric flows''<br />
<br />
We will discuss the clasification of ancient solutions to nonlinear geometric flows. <br />
It is well known that ancient solutions appear as blow up limits at a finite time <br />
singularity of the flow.<br />
Special emphasis will be given to the 2-dimensional Ricci flow.<br />
In this case we will show that ancient compact solution<br />
is either the Einstein (trivial) or one of the King-Rosenau solutions. <br />
<br />
===Maria Gualdani (UT Austin)===<br />
''A nonlinear diffusion model in mean-field games''<br />
<br />
We present an overview of mean-field games theory and show <br />
recent results on a free boundary value problem, which models <br />
price formation dynamics. <br />
In such model, the price is formed through a game among infinite number <br />
of agents. <br />
Existence and regularity results, as well as linear stability, will be shown.<br />
<br />
===Hiroshi Matano (Tokyo University)===<br />
''Traveling waves in a sawtoothed cylinder and their homogenization limit''<br />
<br />
My talk is concerned with a curvature-dependent motion of plane <br />
curves in a two-dimensional cylinder with spatially undulating<br />
boundary. In other words, the boundary has many bumps and we <br />
assume that the bumps are aligned in a spatially recurrent manner.<br />
<br />
The goal is to study how the average speed of the traveling wave <br />
depends on the geometry of the domain boundary. More specifically, <br />
we consider the homogenization problem as the boundary undulation <br />
becomes finer and finer, and determine the homogenization limit <br />
of the average speed and the limit profile of the traveling waves. <br />
Quite surprisingly, this homogenized speed depends only on the <br />
maximal opening angles of the domain boundary and no other <br />
geometrical features are relevant. <br />
<br />
Next we consider the special case where the boundary undulation <br />
is quasi-periodic with ''m'' independent frequencies. We show that <br />
the rate of convergence to the homogenization limit depends on<br />
this number ''m''.<br />
<br />
This is joint work with Bendong Lou and Ken-Ichi Nakamura.<br />
<br />
===Ian Tice (Brown University)===<br />
''Global well-posedness and decay for the viscous surface wave<br />
problem without surface tension''<br />
<br />
We study the incompressible, gravity-driven Navier-Stokes<br />
equations in three dimensional domains with free upper boundaries and<br />
fixed lower boundaries, in both the horizontally periodic and<br />
non-periodic settings. The effect of surface tension is not included.<br />
We employ a novel two-tier nonlinear energy method that couples the<br />
boundedness of certain high-regularity norms to the algebraic decay of<br />
lower-regularity norms. The algebraic decay allows us to balance the<br />
growth of the highest order derivatives of the free surface function,<br />
which then allows us to derive a priori estimates for solutions. We<br />
then prove local well-posedness in our energy space, which yields global<br />
well-posedness and decay. The novel LWP theory is established through<br />
the study of the linear Stokes problem in moving domains. This is joint<br />
work with Yan Guo.<br />
<br />
<br />
===Hoai Minh Nguyen (NYU-Courant Institute)===<br />
''Cloaking via change of variables for the Helmholtz equation''<br />
<br />
A region of space is cloaked for a class of measurements if observers<br />
are not only unaware of its contents, but also unaware of the presence<br />
of the cloak using such measurements. One approach to cloaking is the<br />
change of variables scheme introduced by Greenleaf, Lassas, and<br />
Uhlmann for electrical impedance tomography and by Pendry, Schurig,<br />
and Smith for the Maxwell equation. They used a singular change of<br />
variables which blows up a point into the cloaked region. To avoid<br />
this singularity, various regularized schemes have been proposed. In<br />
this talk I present results related to cloaking via change of<br />
variables for the Helmholtz equation using the natural regularized<br />
scheme introduced by Kohn, Shen, Vogelius, and Weintein, where the<br />
authors used a transformation which blows up a small ball instead of a<br />
point into the cloaked region. I will discuss the degree of<br />
invisibility for a finite range or the full range of frequencies, and<br />
the possibility of achieving perfect cloaking. If time permits, I will<br />
mention some results related to the wave equation.<br />
<br />
===Bing Wang (Princeton)===<br />
''The Kaehler Ricci flow on Fano manifold ''<br />
<br />
We show the convergence of the Kaehler Ricci flow on every 2-dimensional Fano manifold which admits big <math>\alpha_{\nu, 1}</math><br />
or <math>\alpha_{\nu, 2}</math> (Tian's invariants). Our method also works for 2-dimensional Fano orbifolds.<br />
Since Tian's invariants can be calculated by algebraic geometry method, our convergence theorem implies that one can find new Kaehler Einstein metrics<br />
on orbifolds by calculating Tian's invariants.<br />
An essential part of the proof is to confirm the Hamilton-Tian conjecture in complex dimension 2.<br />
<br />
===Francois Hamel (Marseille)===<br />
''Optimization of eigenvalues of non-symmetric elliptic operators''<br />
<br />
The talk is concerned with various optimization results for the principal eigenvalues of general second-order elliptic operators in divergence form with Dirichlet boundary condition in bounded domains of <math>R^n</math>. To each operator in a given domain, we can associate a radially symmetric operator in the ball with the same Lebesgue measure, with a smaller principal eigenvalue. The constraints on the coefficients are of integral, pointwise or geometric types. In particular, we generalize the Rayleigh-Faber-Krahn inequality for the principal eigenvalue of the Dirichlet Laplacian. The proofs use a new symmetrization technique, different from the Schwarz symmetrization. This talk is based on a joint work with N. Nadirashvili and E. Russ.<br />
<br />
===Juraj Foldes (Vanderbilt)===<br />
''Symmetry properties of parabolic problems and their applications''<br />
<br />
Positive solutions of nonlinear parabolic problems can have a very complex behavior. However, assuming certain symmetry conditions, it is possible to prove that the solutions converge to the space of symmetric functions. We show that this property is 'stable'; more specifically if the symmetry conditions are replaced by asymptotically symmetric ones, the solutions still approach the space of symmetric functions. As an application, we show new results on convergence of solutions to a single equilibrium.<br />
<br />
===Alexey Cheskidov (UIC)===<br />
''Navier-Stokes and Euler equations: a unified approach to the problem of blow-up''<br />
<br />
The problems of blow-up for Navier-Stokes and Euler equations <br />
have been extensively studied for decades using different techniques. <br />
Motivated by Kolmogorov's theory of turbulence, we present a new unified <br />
approach to the blow-up problem for the equations of incompressible <br />
fluid motion. In particular, we present a new regularity criterion which <br />
is weaker than the Beale-Kato-Majda condition in the inviscid case, and <br />
weaker than every Ladyzhenskaya-Prodi-Serrin condition in the viscous case.<br />
<br />
===Takis Souganidis (Chicago)===<br />
''Stochastic homogenization of the G-equation''<br />
<br />
The G-equation is a Hamilton-Jacobi equation, of level-set-type, which is used as a model in turbulent combustion. In the lecture I will present recent joint work with Pierre Cardaliaguet about the homogenization of the G-equation set in random media, when the problem is not coercive and, hence, falls outside the scope of the theory of stochastic homogenization.<br />
<br />
===Scott Armstrong (UW-Madison)===<br />
''Partial regularity for fully nonlinear elliptic equations''<br />
<br />
I will present some regularity results for (nonconvex) fully nonlinear equations. Such equations do not possess smooth solutions, but in joint work with Silvestre and Smart we show that the Hausdorff dimension of the singular set is less than the ambient dimension. Using an argument with a similar flavor, we prove (jointly with Silvestre) a unique continuation result for such equations.<br />
<br />
===Russell Schwab (Carnegie Mellon)===<br />
''On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)''<br />
<br />
Despite much recent (and not so recent) attention to solutions of integro-differential equations of elliptic type, it is surprising that a fundamental result such as a comparison theorem which can deal with only measure theoretic norms of the right hand side of the equation (L-n and L-infinity) has gone unexplored. For the case of second order equations this result is known as the Aleksandrov-Bakelman-Pucci estimate (and dates back to circa 1960s), which says that for supersolutions of uniformly elliptic equation Lu=f, the supremum of u is controlled by the L-n norm of f (n being the underlying dimension of the domain). We discuss extensions of this estimate to fully nonlinear integro-differential equations and present a recent result in this direction. (Joint with Nestor Guillen, available at arXiv:1101.0279v3 [math.AP])<br />
<br />
===Valentin Ovsienko (University of Lyon)===<br />
''The pentagram map and generalized friezes of Coxeter''<br />
<br />
The pentagram map is a discrete integrable system on the moduli space of n-gons in the projective plane (which is a close relative of the moduli space of genus 0 curves with n marked points). The most interesting properties of the pentagram map is its relations to the theory of cluster algebras and to the classical integrable systems (such as the Boussinesq equation). I will talk of the recent results proving the integrability as well as of the algebraic and arithmetic properties of the pentagram map.<br />
In particular, I will introduce the space of 2-frieze patterns generalizing that of the classical Coxeter friezes and define the structure of cluster manifold on this space. The talk is based on joint works with Sophie Morier-Genoud, Richard Schwartz and Serge Tabachnikov.<br />
<br />
<br />
===Adrian Tudorascu (West Virginia University)===<br />
''Weak Lagrangian solutions for the Semi-Geostrophic system in physical space''<br />
<br />
Proposed as a simplification for the Boussinesq system in a special regime, the Semi-Geostrophic (SG) system is used by metereologists to model how fronts arise in large scale weather patterns. In spite of significant progress achieved in the analysis of the SG in dual space (i.e. the system obtained from the SG by a special change of variables), there are no existence results on the SG in physical space. We shall argue that weak (Eulerian) solutions for the Semi-Geostrophic system in physical space exhibiting some mild regularity in time cannot yield point masses in the dual space. However, such solutions are physically relevant to the model. Thus, we shall discuss a natural generalization of Cullen & Feldman's weak Lagrangian solutions in the physical space to include the possibility of singular measures in dual space. We have proved existence of such solutions in the case of discrete measures in dual space. The talk is based on joint work with M. Feldman.<br />
<br />
===James Nolen (Duke)===<br />
''Normal approximation for a random elliptic PDE''<br />
<br />
I will talk about solutions to an elliptic PDE with conductivity coefficient that varies randomly with respect to the spatial variable. It has been known for some time that homogenization may occur when the coefficients are scaled suitably. Less is known about fluctuations of the solution around its mean behavior. This talk is about the energy dissipation rate, which is a quadratic functional of the solution and a bulk property of the random material. For a finite random sample of the material, this quantity is random. In the limit of large sample size it converges to a deterministic constant. I will describe a central limit theorem: the probability law of the energy dissipation rate is very close to that of a normal random variable having the same mean and variance. I'll give an error estimate for this approximation in total variation.<br />
<br />
===Betsy Stovall (UCLA)===<br />
<br />
We will discuss recent work concerning the cubic Klein--Gordon equation<br />
u_{tt} - \Delta u + u \pm u^3 = 0<br />
in two space dimensions with real valued initial data in the energy space, u(0) in H^1, u_t(0) in L^2. We show that in the defocusing case, solutions<br />
are global and have finite L^4 norm (in space and time). In the focusing case, we characterize the dichotomy between such behavior and finite time<br />
blowup for initial data having energy less than that of the ground state. In this talk, we will pay particular attention to connections with certain<br />
questions arising in harmonic analysis. <br />
<br />
This is joint work with Rowan Killip and Monica Visan.<br />
<br />
===Charles Smart (MIT)===<br />
''Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian''<br />
<br />
A classical theorem of Kirszbraun states that any Lipschitz function $f : A \to \R^m$ defined on a set $A \subseteq \R^n$ can be extended to all of $\R^n$ without increasing the Lipschitz constant. The search for a canonical such extension leads one to the notion of optimal Lipschitz extension. I will discuss joint work with Evans on the regularity of optimal Lipschitz extensions in the scalar $m = 1$ case and joint work with Sheffield on the vector valued $m > 1$ case.<br />
<br />
<br />
===Yao Yao (UCLA)===<br />
''Degenerate diffusion with nonlocal aggregation: behavior of solutions''<br />
<br />
The Patlak-Keller-Segel (PKS) equation models the collective motion of<br />
cells which are attracted by a self-emitted chemical substance. While the<br />
global well-posedness and finite-time blow up criteria are well known, the<br />
asymptotic behaviors of solutions are not completely clear. In this talk I<br />
will present some results on the asymptotic behavior of solutions when<br />
there is global existence. The key tools used in the paper are<br />
maximum-principle type arguments as well as estimates on mass concentration<br />
of solutions. This is a joint work with Inwon Kim.<br />
<br />
===Xuan Hien Nguyen (Iowa State)===<br />
<br />
''Gluing constructions for solitons and self-shrinkers under mean curvature flow''<br />
<br />
In the 1990s, Kapouleas and Traizet constructed new examples of minimal surfaces by desingularizing the intersection of existing ones with Scherk surfaces. Using this idea, one can find new examples of self-translating solutions for the mean curvature flow asymptotic at infinity to a finite family of grim reaper cylinders in general position. Recently, it has been shown that it is possible to desingularize the intersection of a sphere and a plane to obtain a family of self-shrinkers under mean curvature flow. I will discuss the main steps and difficulties for these gluing constructions, as well as open problems.<br />
<br />
===Nestor Guillen (UCLA)===<br />
<br />
We consider the Monge-Kantorovich problem, which consists in<br />
transporting a given measure into another "target" measure in a way<br />
that minimizes the total cost of moving each unit of mass to its new<br />
location. When the transport cost is given by the square of the<br />
distance between two points, the optimal map is given by a convex<br />
potential which solves the Monge-Ampère equation, in general, the<br />
solution is given by what is called a c-convex potential. In recent<br />
work with Jun Kitagawa, we prove local Holder estimates of optimal<br />
transport maps for more general cost functions satisfying a<br />
"synthetic" MTW condition, in particular, the proof does not really<br />
use the C^4 assumption made in all previous works. A similar result<br />
was recently obtained by Figalli, Kim and McCann using different<br />
methods and assuming strict convexity of the target.<br />
<br />
===Charles Smart (MIT)===<br />
<br />
''PDE methods for the Abelian sandpile''<br />
<br />
Abstract: The Abelian sandpile growth model is a deterministic<br />
diffusion process for chips placed on the $d$-dimensional integer<br />
lattice. One of the most striking features of the sandpile is that it<br />
appears to produce terminal configurations converging to a peculiar<br />
lattice. One of the most striking features of the sandpile is that it<br />
appears to produce terminal configurations converging to a peculiar<br />
fractal limit when begun from increasingly large stacks of chips at<br />
the origin. This behavior defied explanation for many years until<br />
viscosity solution theory offered a new perspective. This is joint<br />
work with Lionel Levine and Wesley Pegden.<br />
<br />
===Vlad Vicol (University of Chicago)===<br />
<br />
Title: Shape dependent maximum principles and applications<br />
<br />
Abstract: We present a non-linear lower bound for the fractional Laplacian, when<br />
evaluated at extrema of a function. Applications to the global well-posedness of active<br />
scalar equations arising in fluid dynamics are discussed. This is joint work with P.<br />
Constantin.<br />
<br />
<br />
===Jiahong Wu (Oklahoma State)===<br />
<br />
"The 2D Boussinesq equations with partial dissipation"<br />
<br />
The Boussinesq equations concerned here model geophysical flows such<br />
as atmospheric fronts and ocean circulations. Mathematically the 2D Boussinesq<br />
equations serve as a lower-dimensional model of the 3D hydrodynamics<br />
equations. In fact, the 2D Boussinesq equations retain some key features<br />
of the 3D Euler and the Navier-Stokes equations such as the vortex stretching<br />
mechanism. The global regularity problem on the 2D Boussinesq equations<br />
with partial dissipation has attracted considerable attention in the last few years.<br />
In this talk we will summarize recent results on various cases of partial dissipation,<br />
present the work of Cao and Wu on the 2D Boussinesq equations with vertical<br />
dissipation and vertical thermal diffusion, and explain the work of Chae and Wu on<br />
the critical Boussinesq equations with a logarithmically singular velocity.<br />
<br />
<br />
===Joana Oliveira dos Santos Amorim (Universit\'e Paris Dauphine)===<br />
<br />
"A geometric look on Aubry-Mather theory and a theorem of Birkhoff"<br />
<br />
Given a Tonelli Hamiltonian $H:T^*M \lto \Rm$ in the cotangent bundle of a compact manifold $M$, <br />
we can study its dynamic using the Aubry and Ma\~n\'e sets defined by Mather. <br />
In this talk we will explain their importance and give a new geometric definition <br />
which allows us to understand their property of symplectic invariance. <br />
Moreover, using this geometric definition, we will show that an exact <br />
Lipchitz Lagrangian manifold isotopic to a graph which is invariant <br />
by the flow of a Tonelli Hamiltonian is itself a graph. <br />
This result, in its smooth form, was a conjecture of Birkhoff.<br />
<br />
<br />
===Gui-Qiang Chen (Oxford) ===<br />
<br />
"Nonlinear Partial Differential Equations of Mixed Type"<br />
<br />
Many nonlinear partial differential equations arising in mechanics and geometry naturally are of mixed hyperbolic-elliptic type. The solution of some longstanding fundamental problems in these areas greatly requires a deep understanding of such nonlinear partial differential equations of mixed type. Important examples include<br />
shock reflection-diffraction problems in fluid mechanics (the Euler equations), isometric embedding problems in in differential geometry (the Gauss-Codazzi-Ricci equations),<br />
among many others. In this talk we will present natural connections of nonlinear partial differential equations with these longstanding problems and will discuss some recent developments in the analysis of these nonlinear equations through the examples with emphasis on identifying/developing mathematical approaches, ideas, and techniques to deal with the mixed-type problems. Further trends, perspectives, and open problems in this direction will also be addressed.<br />
This talk will be based mainly on the joint work correspondingly with Mikhail Feldman, Marshall Slemrod, as well as Myoungjean Bae and Dehua Wang. <br />
<br />
<br />
===Jacob Glenn-Levin (UT Austin)===<br />
<br />
We consider the Boussinesq equations, which may be thought of as inhomogeneous, <br />
incompressible Euler equations, where the inhomogeneous term is a scalar quantity, <br />
typically density or temperature, governed by a convection-diffusion equation. I will <br />
discuss local- and global-in-time well-posedness results for the incompressible 2D <br />
Boussinesq equations, assuming the density equation has nonzero diffusion and that the <br />
initial data belongs in a Besov-type space.<br />
<br />
===Bing Wang (UW Madison)===<br />
''On the regularity of limit space''<br />
<br />
This is a joint work with Gang Tian. <br />
In this talk, we will discuss how to improve regularity of the limit space by Ricci flow.<br />
We study the structure of the limit space of a sequence of almost Einstein<br />
manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such<br />
manifolds are the initial manifolds of some normalized Ricci flows whose scalar<br />
curvatures are almost constants over space-time in the L1-sense, Ricci curvatures<br />
are bounded from below at the initial time. Under the non-collapsed condition,<br />
we show that the limit space of a sequence of almost Einstein manifolds has most<br />
properties which is known for the limit space of Einstein manifolds. As applications,<br />
we can apply our structure results to study the properties of K¨ahler manifolds.<br />
<br />
<br />
===Peter Polacik (University of Minnesota)===<br />
'' Exponential separation between positive and sign-changing solutions and its applications''<br />
<br />
In linear nonautonomous second-order parabolic equations, the exponential separation refers to the exponential decay of any sign-changing solution relative to any positive solution. In this lecture, after summarizing key results on exponential separation, we show how it can be effectively used in studies of some nonlinear parabolic problems.<br />
<br />
<br />
<br />
===Kyudong Choi (UW Madison)===<br />
''Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations''<br />
<br />
We study weak solutions of the 3D Navier-Stokes equations with L^2 initial data. We <br />
prove that k-th derivative of weak solutions is locally integrable in space-time for any <br />
real k such that 1 < k < 3. Up to now, only the second derivative was known to be locally <br />
integrable by standard parabolic regularization. We also present sharp estimates of those <br />
quantities in weak-L^{4/(k+1)} locally. These estimates depend only on the L^2 norm of <br />
the initial data and on the domain of integration. Moreover, they are valid even for k >= <br />
3 as long as we have a smooth solution. The proof uses a standard approximation of <br />
Navier-Stokes from Leray and a blow-up techniques. The local study is based on De Giorgi <br />
techniques with a new pressure decomposition. To handle the non-locality of fractional <br />
Laplacians, Hardy space and Maximal functions are introduced. This is joint work with A. <br />
Vasseur.<br />
<br />
<br />
===Yao Yao (UW Madison)===<br />
''Confinement for nonlocal interaction equation with repulsive-attractive kernels''<br />
<br />
In this talk, we consider sets of particles interacting pairwise via a potential W, which is repulsive at short distances, but attractive at long distances. The main question we consider is whether an initially compactly supported configuration remains compactly supported for all times, regardless of the number of particles. We improved the sufficient conditions on the potential W for the above confinement property to hold. This is a joint work with Jose Carrillo and Daniel Balague.<br />
<br />
<br />
<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Transonic shocks for Euler-Poisson system and related problems''<br />
<br />
Abstract: Euler-Poisson system models various physical phenomena<br />
including the propagation of electrons in submicron semiconductor<br />
devices and plasmas, and the biological transport of ions for channel<br />
proteins. I will explain difference between Euler system and<br />
Euler-Poisson system and mathematical difficulties arising due to this<br />
difference. And, recent results about subsonic flow and transonic flow<br />
for Euler-Poisson system will be presented. This talk is based on<br />
collaboration with Ben Duan and Chunjing Xie.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Previous_PDE/GA_seminars&diff=17673Previous PDE/GA seminars2019-08-26T16:54:53Z<p>Ckim: </p>
<hr />
<div>[[PDE_Geometric_Analysis_seminar|Click here]] for the current PDE and Geometric Analysis seminar schedule.<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
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{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | Stability of vacuum for the Landau equation with moderately soft potentials ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria. ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | Gradient estimate for complex Monge-Ampere equations ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | Speeds and Homogenization for Reaction-Diffusion Equations in Random Media ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Spring 2018 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|January 29, '''3-3:50PM, B341 VV.'''<br />
| Dan Knopf (UT Austin)<br />
|[[#Dan Knopf | Non-K&auml;hler Ricci flow singularities that converge to K&auml;hler-Ricci solitons]]<br />
| Angenent<br />
|- <br />
|February 5, '''3-3:50PM, B341 VV.'''<br />
| Andreas Seeger (UW)<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows ]]<br />
| Kim & Tran<br />
|- <br />
|February 12<br />
| Sam Krupa (UT-Austin)<br />
|[[#Sam Krupa | Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case ]]<br />
| Lee <br />
|- <br />
|February 19<br />
| Maja Taskovic (UPenn)<br />
|[[#Maja Taskovic | Exponential tails for the non-cutoff Boltzmann equation ]]<br />
| Kim<br />
|- <br />
|February 26<br />
| Ashish Kumar Pandey (UIUC)<br />
|[[# | Instabilities in shallow water wave models ]]<br />
| Kim & Lee<br />
|- <br />
|March 5<br />
| Khai Nguyen (NCSU)<br />
|[[#Khai Nguyen | Burgers Equation with Some Nonlocal Sources ]]<br />
| Tran<br />
|- <br />
|March 12<br />
| Hongwei Gao (UCLA)<br />
|[[#Hongwei Gao | Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations ]]<br />
| Tran<br />
|- <br />
|March 19<br />
| Huy Nguyen (Princeton)<br />
|[[#Huy Nguyen | Compressible fluids and active potentials ]]<br />
| Lee<br />
|-<br />
|March 26<br />
| <br />
|[[# | Spring recess (Mar 24-Apr 1, 2018) ]]<br />
| <br />
|-<br />
|April 2<br />
| In-Jee Jeong (Princeton)<br />
|[[#In-Jee Jeong | Singularity formation for the 3D axisymmetric Euler equations ]]<br />
| Kim<br />
|- <br />
|April 9<br />
| Jeff Calder (Minnesota)<br />
|[[#Jeff Calder | Nonlinear PDE continuum limits in data science and machine learning ]]<br />
| Tran<br />
|- <br />
|April 21-22 (Saturday-Sunday)<br />
| [https://sites.google.com/view/81stmidwestpdeseminar/home Midwest PDE seminar]<br />
|[[#Midwest PDE seminar | ]]<br />
| Angenent, Feldman, Kim, Tran.<br />
|- <br />
|April 25 (Wednesday)<br />
| Hitoshi Ishii (Wasow lecture)<br />
|[[#Hitoshi Ishii | Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory]]<br />
| Tran.<br />
|}<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2017 ==<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|September 11<br />
|Mihaela Ifrim (UW)<br />
|[[#Mihaela Ifrim| Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation]]<br />
| Kim & Tran<br />
|- <br />
|September 18<br />
|Longjie Zhang (University of Tokyo) <br />
|[[#Longjie Zhang | On curvature flow with driving force starting as singular initial curve in the plane]]<br />
| Angenent<br />
|- <br />
|September 22,<br />
VV 9th floor hall, 4:00pm<br />
|Jaeyoung Byeon (KAIST) <br />
|[[#Jaeyoung Byeon| Colloquium: Patterns formation for elliptic systems with large interaction forces]]<br />
| Rabinowitz <br />
|- <br />
|September 25<br />
| Tuoc Phan (UTK)<br />
|[[#Tuoc Phan | Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application]]<br />
| Tran<br />
|- <br />
|September 26, <br />
VV B139 4:00pm<br />
| Hiroyoshi Mitake (Hiroshima University)<br />
|[[#Hiroyoshi Mitake | Joint Analysis/PDE seminar: Derivation of multi-layered interface system and its application]]<br />
| Tran<br />
|- <br />
|September 29,<br />
VV901 2:25pm<br />
| Dongnam Ko (CMU & SNU)<br />
|[[#Dongnam Ko | a joint seminar with ACMS: On the emergence of local flocking phenomena in Cucker-Smale ensembles ]]<br />
| Shi Jin & Kim<br />
|- <br />
|October 2<br />
| No seminar due to a KI-Net conference<br />
|<br />
|<br />
|- <br />
|October 9<br />
| Sameer Iyer (Brown University)<br />
|[[#Sameer Iyer | Global-in-x Steady Prandtl Expansion over a Moving Boundary ]]<br />
| Kim<br />
|- <br />
|October 16<br />
| Jingrui Cheng (UW)<br />
|[[#Jingrui Cheng | A 1-D semigeostrophic model with moist convection ]]<br />
| Kim & Tran<br />
|- <br />
|October 23<br />
| Donghyun Lee (UW)<br />
|[[#Donghyun Lee | The Vlasov-Poisson-Boltzmann system in bounded domains ]]<br />
| Kim & Tran<br />
|- <br />
|October 30<br />
| Myoungjean Bae (POSTECH)<br />
|[[#Myoungjean Bae | 3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system ]]<br />
| Feldman<br />
|- <br />
|November 6<br />
| Jingchen Hu (USTC and UW)<br />
|[[#Jingchen Hu | Shock Reflection and Diffraction Problem with Potential Flow Equation ]]<br />
| Kim & Tran <br />
|- <br />
|November 20<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization and Harnack inequality for a degenerate discrete nondivergence form random operator ]]<br />
| Kim & Tran <br />
|- <br />
|November 27 (special time: 3-4PM)<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse problems for Maxwell's equations and its application ]]<br />
| Li<br />
|- <br />
|December 4<br />
| Norbert Pozar (Kanazawa University)<br />
|[[#Norbert Pozar | Viscosity solutions for the crystalline mean curvature flow ]]<br />
| Tran<br />
|- <br />
|December 5 (Tuesday), Colloquium<br />
| Ryan Hynd (U Penn)<br />
|[[#Ryan Hynd | TBD ]]<br />
| <br />
|- <br />
|December 11 (Monday), Colloquium<br />
| Connor Mooney (ETH Zurich)<br />
|[[#Connor Mooney| TBD ]]<br />
| <br />
|}<br />
<br />
<br />
<br />
<br />
= PDE GA Seminar Schedule Spring 2017 =<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|-<br />
|January 23<br>Special time and location:<br> 3-3:50pm, B325 Van Vleck<br />
| Sigurd Angenent (UW)<br />
|[[#Sigurd Angenent | Ancient convex solutions to Mean Curvature Flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|January 30<br />
| Serguei Denissov (UW)<br />
|[[#Serguei Denissov | Instability in 2D Euler equation of incompressible inviscid fluid]]<br />
| Kim & Tran<br />
|- <br />
<br />
<br />
|-<br />
|February 6 - Wasow lecture<br />
| Benoit Perthame (University of Paris VI)<br />
|[[#| ]]<br />
| Jin<br />
|- <br />
<br />
<br />
|-<br />
|February 13<br />
| Bing Wang (UW)<br />
|[[#Bing Wang | The extension problem of the mean curvature flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|February 20<br />
| Eric Baer (UW)<br />
|[[#Eric Baer | Isoperimetric sets inside almost-convex cones]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|February 27<br />
| Ben Seeger (University of Chicago)<br />
|[[#Ben Seeger | Homogenization of pathwise Hamilton-Jacobi equations ]]<br />
| Tran<br />
|- <br />
<br />
|-<br />
|March 7 - Mathematics Department Distinguished Lecture<br />
| Roger Temam (Indiana University) <br />
|[[#Roger Temam | On the mathematical modeling of the humid atmosphere]]<br />
| Smith <br />
|- <br />
<br />
<br />
|-<br />
|March 8 - Analysis/Applied math/PDE seminar<br />
| Roger Temam (Indiana University) <br />
|[[#Roger Temam | Weak solutions of the Shigesada-Kawasaki-Teramoto system ]]<br />
| Smith<br />
|-<br />
<br />
|-<br />
|March 13<br />
| Sona Akopian (UT-Austin)<br />
|[[#Sona Akopian | Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.]]<br />
| Kim<br />
<br />
|-<br />
|March 27 - Analysis/PDE seminar<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | Mean-Field Limits for Ginzburg-Landau vortices ]]<br />
| Tran<br />
<br />
|-<br />
|March 29 - Wasow lecture<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | Microscopic description of Coulomb-type systems ]]<br />
| <br />
<br />
<br />
|-<br />
|March 30 <br>Special day (Thursday) and location:<br> B139 Van Vleck<br />
| Gui-Qiang Chen (Oxford)<br />
|[[#Gui-Qiang Chen | Supersonic Flow onto Solid Wedges,<br />
Multidimensional Shock Waves and Free Boundary Problems ]]<br />
| Feldman<br />
<br />
<br />
<br />
|-<br />
|April 3<br />
| Zhenfu Wang (Maryland)<br />
|[[#Zhenfu Wang | Mean field limit for stochastic particle systems with singular forces]]<br />
| Kim<br />
<br />
|-<br />
|April 10<br />
| Andrei Tarfulea (Chicago)<br />
|[[#Andrei Tarfulea | Improved estimates for thermal fluid equations]]<br />
| Baer<br />
<br />
|-<br />
|April 17 <br>Special time and location:<br> 3-3:50pm, B219 Van Vleck<br />
| Siao-Hao Guo (Rutgers)<br />
|[[# Siao-Hao Guo | Analysis of Velázquez's solution to the mean curvature flow with a type II singularity]]<br />
| Lu Wang<br />
<br />
<br />
|-<br />
|April 24<br />
| Jianfeng Lu (Duke)<br />
|[[#Jianfeng Lu | Evolution of crystal surfaces: from mesoscopic to continuum models]]<br />
| Li<br />
<br />
|-<br />
|April 25- joint Analysis/PDE seminar<br />
| Chris Henderson (Chicago)<br />
|[[#Chris Henderson | A local-in-time Harnack inequality and applications to reaction-diffusion equations<br />
<br />
]]<br />
| Lin<br />
<br />
|-<br />
|May 1st(Special time: 4:00-5:00pm)<br />
| Jeffrey Streets (UC-Irvine)<br />
|[[#Jeffrey Streets | Generalized Kahler Ricci flow and a generalized Calabi conjecture]]<br />
| Bing Wang<br />
|}<br />
<br />
<br />
<br />
= PDE GA Seminar Schedule Fall 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 12<br />
| Daniel Spirn (U of Minnesota)<br />
|[[#Daniel Spirn | Dipole Trajectories in Bose-Einstein Condensates ]]<br />
| Kim<br />
|-<br />
|September 19<br />
| Donghyun Lee (UW-Madison)<br />
|[[#Donghyun Lee | The Boltzmann equation with specular boundary condition in convex domains ]]<br />
| Feldman<br />
|-<br />
|September 26<br />
| Kevin Zumbrun (Indiana)<br />
|[[#Kevin Zumbrun | A Stable Manifold Theorem for a class of degenerate evolution equations ]]<br />
| Kim <br />
|-<br />
|October 3<br />
| Will Feldman (UChicago )<br />
|[[#Will Feldman | Liquid Drops on a Rough Surface ]]<br />
| Lin & Tran<br />
|-<br />
|October 10<br />
| Ryan Hynd (UPenn)<br />
|[[#Ryan Hynd | Extremal functions for Morrey’s inequality in convex domains ]]<br />
| Feldman<br />
|-<br />
|October 17<br />
| Gung-Min Gie (Louisville)<br />
|[[#Gung-Min Gie | Boundary layer analysis of some incompressible flows ]]<br />
| Kim<br />
|-<br />
|October 24<br />
| Tau Shean Lim (UW Madison)<br />
|[[#Tau Shean Lim | Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators ]]<br />
| Kim & Tran<br />
|-<br />
|October 31 ('''Special time and room''': B313VV, 3PM-4PM)<br />
| Tarek Elgindi ( Princeton)<br />
|[[#Tarek Elgindi | Propagation of Singularities in Incompressible Fluids ]]<br />
| Lee & Kim<br />
|-<br />
|November 7<br />
| Adrian Tudorascu (West Virginia)<br />
|[[#Adrian Tudorascu | Hamilton-Jacobi equations in the Wasserstein space of probability measures ]]<br />
| Feldman<br />
|-<br />
|November 14<br />
| Alexis Vasseur ( UT-Austin)<br />
|[[#Alexis Vasseur | Compressible Navier-Stokes equations with degenerate viscosities ]]<br />
| Feldman<br />
|-<br />
|November 21<br />
| Minh-Binh Tran (UW Madison )<br />
|[[#Minh-Binh Tran | Quantum Kinetic Problems ]]<br />
| Hung Tran<br />
|-<br />
|November 28<br />
| David Kaspar (Brown)<br />
|[[#David Kaspar | Kinetics of shock clustering ]]<br />
|Tran<br />
|-<br />
|December 5 ('''Special time and room''': 3PM-4PM, B313VV)<br />
| Brian Weber (University of Pennsylvania)<br />
|[[#Brian Weber | Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds ]]<br />
|Bing Wang<br />
|-<br />
|December 12<br />
| (no seminar)<br />
|[[# | ]]<br />
| <br />
|}<br />
<br />
<br />
<br />
= Seminar Schedule Spring 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 25<br />
||Tianling Jin (HKUST and Caltech)<br />
|[[#Tianling Jin | Holder gradient estimates for parabolic homogeneous p-Laplacian equations ]]<br />
| Zlatos<br />
|-<br />
|February 1<br />
|Russell Schwab (Michigan State University)<br />
|[[#Russell Schwab | Neumann homogenization via integro-differential methods ]]<br />
| Lin<br />
|-<br />
|February 8 <br />
|Jingrui Cheng (UW Madison)<br />
|[[#Jingrui Cheng | Semi-geostrophic system with variable Coriolis parameter ]]<br />
| Tran & Kim<br />
|-<br />
|February 15 <br />
|Paul Rabinowitz (UW Madison)<br />
|[[#Paul Rabinowitz | On A Double Well Potential System ]]<br />
| Tran & Kim<br />
|-<br />
|February 22 <br />
|Hong Zhang (Brown)<br />
|[[#Hong Zhang | On an elliptic equation arising from composite material ]]<br />
| Kim<br />
|-<br />
|February 29<br />
|Aaron Yip (Purdue university) <br />
|[[#Aaron Yip | Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media ]]<br />
| Tran<br />
|-<br />
|March 7<br />
|Hiroyoshi Mitake (Hiroshima university) <br />
||[[#Hiroyoshi Mitake | Selection problem for fully nonlinear equations]]<br />
| Tran<br />
|-<br />
|March 15<br />
|Nestor Guillen (UMass Amherst)<br />
|[[#Nestor Guillen | Min-max formulas for integro-differential equations and applications ]]<br />
| Lin<br />
|-<br />
|March 21 (Spring Break)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 28<br />
|Ryan Denlinger (Courant Institute)<br />
|[[#Ryan Denlinger | The propagation of chaos for a rarefied gas of hard spheres in vacuum ]]<br />
| Lee<br />
|-<br />
|April 4<br />
| No seminar<br />
||[[# | ]]<br />
| <br />
|-<br />
|April 11<br />
|Misha Feldman (UW)<br />
|[[#Misha Feldman | Shock reflection, free boundary problems and degenerate elliptic equations ]]<br />
|<br />
|-<br />
|April 14: 2:25 PM in VV 901-Joint with Probability Seminar<br />
|Jessica Lin (UW-Madison)<br />
|[[#Jessica Lin | Optimal Quantitative Error Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form ]]<br />
|-<br />
|April 18<br />
|Sergey Bolotin (UW-Madison)<br />
|[[#Sergey Bolotin | Degenerate billiards]]<br />
|<br />
|-<br />
|April 21-24, KI-Net conference: Boundary Value Problems and Multiscale Coupling Methods for Kinetic Equations<br />
|Link: http://www.ki-net.umd.edu/content/conf?event_id=493<br />
|-<br />
|April 25<br />
| Moon-Jin Kang (UT-Austin)<br />
|[[#Moon-Jin Kang | On contraction of large perturbation of shock waves, and inviscid limit problems ]]<br />
| Kim<br />
|-<br />
|Tuesday, May 3, 4:00 p.m., in B139 (Joint Analysis-PDE seminar )<br />
|Stanley Snelson (University of Chicago) <br />
|[[# | ]]<br />
| Seeger & Tran.<br />
|-<br />
|May 16-20, Conference in Harmonic Analysis in Honor of Michael Christ<br />
|Link: https://www.math.wisc.edu/ha_2016/<br />
|-<br />
|}<br />
<br />
<br />
<br />
= Seminar Schedule Fall 2015 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|September 7 (Labor Day)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|September 14 (special room: B115)<br />
| Hung Tran (Madison)<br />
|[[#Hung Tran | Some inverse problems in periodic homogenization of Hamilton--Jacobi equations ]]<br />
| <br />
|- <br />
|September 21 (special room: B115) <br />
| Eric Baer (Madison)<br />
||[[#Eric Baer | Optimal function spaces for continuity of the Hessian determinant as a distribution ]]<br />
| <br />
|-<br />
|September 28<br />
| Donghyun Lee (Madison)<br />
|[[#Donghyun Lee | FLUIDS WITH FREE-SURFACE AND VANISHING VISCOSITY LIMIT]]<br />
| <br />
|-<br />
|October 5 <br />
|Hyung-Ju Hwang (Postech & Brown Univ)<br />
|[[#Hyung-Ju Hwang | The Fokker-Planck equation in bounded domains ]]<br />
| Kim<br />
|-<br />
|October 12<br />
| Minh-Binh Tran (Madison)<br />
|[[#Minh-Binh Tran | Nonlinear approximation theory for kinetic equations ]]<br />
| <br />
|-<br />
|October 19<br />
| Bob Jensen (Loyola University Chicago)<br />
||[[#Bob Jensen | Crandall-Lions Viscosity Solutions of Uniformly Elliptic PDEs ]]<br />
| Tran<br />
|-<br />
|October 26<br />
|Luis Silvestre (Chicago)<br />
|[[#Luis Silvestre | A priori estimates for integral equations and the Boltzmann equation ]]<br />
|Kim<br />
|-<br />
|November 2<br />
| Connor Mooney (UT Austin)<br />
|[[#Connor Mooney | Counterexamples to Sobolev regularity for degenerate Monge-Ampere equations ]]<br />
|Lin<br />
|-<br />
|November 9<br />
| Javier Gomez Serrano (Princeton)<br />
||[[#Javier Gomez Serrano | Existence and regularity of rotating global solutions for active scalars ]]<br />
|Zlatos<br />
|-<br />
|November 16<br />
| Yifeng Yu (UC Irvine)<br />
|[[#Yifeng Yu | G-equation in the modeling of flame propagation ]]<br />
| Tran<br />
|-<br />
|November 23<br />
| Nam Le (Indiana)<br />
|[[#Nam Le | Global smoothness of the Monge-Ampere eigenfunctions ]]<br />
|Tran<br />
|-<br />
|November 30<br />
| Qin Li (Madison)<br />
|[[#Qin Li | Kinetic-fluid coupling: transition from the Boltzmann to the Euler ]]<br />
|<br />
|-<br />
|December 7<br />
| Lu Wang (Madison) <br />
||[[#Lu Wang | Asymptotic Geometry of Self-shrinkers ]]<br />
|<br />
|-<br />
|December 14<br />
| Christophe Lacave (Paris 7)<br />
|[[# Christophe Lacave | Well-posedness for 2D Euler in non-smooth domains ]]<br />
| Zlatos<br />
|}<br />
<br />
<br />
<br />
= Seminar Schedule Spring 2015 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 21 (Departmental Colloquium: 4PM, B239) <br />
|Jun Kitagawa (Toronto) <br />
|[[#Jun Kitagawa (Toronto) | Regularity theory for generated Jacobian equations: from optimal transport to geometric optics ]]<br />
|Feldman <br />
|-<br />
|February 9 <br />
|Jessica Lin (Madison)<br />
|[[#Jessica Lin (Madison) | Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations ]]<br />
|Kim<br />
|-<br />
|February 17 (Tuesday) (joint with Analysis Seminar: 4PM, B139)<br />
|Chanwoo Kim (Madison) <br />
|[[#Chanwoo Kim (Madison) | Hydrodynamic limit from the Boltzmann to the Navier-Stokes-Fourier ]]<br />
|Seeger <br />
|-<br />
|February 23 (special time*, '''3PM, B119''') <br />
| Yaguang Wang (Shanghai Jiao Tong)<br />
|[[ #Yaguang Wang | Stability of Three-dimensional Prandtl Boundary Layers ]]<br />
|Jin<br />
|-<br />
|March 2 <br />
|Benoit Pausader (Princeton)<br />
|[[#Benoit Pausader (Princeton) | Global smooth solutions for the Euler-Maxwell problem for electrons in 2 dimensions]]<br />
|Kim<br />
|-<br />
|March 9 <br />
|Haozhao Li (University of Science and Technology of China) <br />
|[[#Haozhao Li|Regularity scales and convergence of the Calabi flow]]<br />
|Wang <br />
|-<br />
|March 16 <br />
| Jennifer Beichman (Madison) <br />
|[[#Jennifer Beichman (Madison) |Nonstandard dispersive estimates and linearized water waves ]]<br />
| Kim<br />
|-<br />
|March 23 <br />
| Ben Fehrman (University of Chicago)<br />
|[[#Ben Fehrman (University of Chicago) | On The Existence of an Invariant Measure for Isotropic Diffusions in Random Environments ]]<br />
| Lin<br />
|-<br />
|March 30 <br />
| Spring recess Mar 28-Apr 5 (S-N)<br />
|[[# | ]]<br />
| <br />
|-<br />
|April 13 <br />
| Sung-Jin Oh (Berkeley)<br />
|[[# Berkeley | Global well-posedness of the energy critical Maxwell-Klein-Gordon equation ]]<br />
| Kim<br />
|-<br />
|April 20 <br />
|Yuan Lou (Ohio State)<br />
|[[#Yuan Lou (Ohio State) | Asymptotic behavior of the smallest eigenvalue of an elliptic operator and its applications to evolution of dispersal]]<br />
|Zlatos<br />
|-<br />
|'''April 28''' (a joint seminar with analysis, '''4:00 p.m B139''')<br />
| Diego Córdoba (ICMAT, Madrid) <br />
|[[# Diego Córdoba |Global existence solutions and geometric properties of the SQG sharp front ]]<br />
| Zlatos<br />
|-<br />
|May 4 <br />
| Vera Hur (UIUC) <br />
|[[# Vera Hur (UIUC) |Instabilities in nonlinear dispersive waves ]]<br />
| Yao<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Fall 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker <br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 15 <br />
|Greg Kuperberg (UC-Davis)<br />
|[[#Greg Kuperberg| ''Cartan-Hadamard and the Little Prince'' ]]<br />
|Viaclovsky<br />
|-<br />
|-<br />
|September 22 (joint with Analysis Seminar)<br />
|Steve Hofmann (U. of Missouri)<br />
|[[#Steve Hofmann| ''Quantitative Rectifiability and Elliptic Equations'']]<br />
|Seeger<br />
|-<br />
|-<br />
|Oct 6th, <br />
|Xiangwen Zhang (Columbia University)<br />
|[[#Xiangwen Zhang |Alexandrov's Uniqueness Theorem for Convex Surfaces ]]<br />
|B.Wang<br />
|-<br />
|-<br />
|October 13 <br />
|Xuwen Chen (Brown University)[http://www.math.brown.edu/~chenxuwen/]<br />
|[[#Xuwen Chen| The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-body Evolution]]<br />
|C.Kim<br />
|-<br />
|-<br />
|October 20 <br />
|Kyudong Choi (UW-Madison)<br />
|[[#Kyudong Choi| Finite time blow up for 1D models for the 3D Axisymmetric Euler Equations the 2D Boussinesq system]]<br />
|C.Kim<br />
|-<br />
|-<br />
|October 27<br />
|Chanwoo Kim (UW-Madison)<br />
|[[#Chanwoo Kim|<br />
BV-Regularity of the Boltzmann Equation in Non-Convex Domains]]<br />
|Local<br />
|-<br />
|-<br />
|November 3 <br />
|Myoungjean Bae (POSTECH)<br />
|[[#Myoungjean Bae| Recent progress on study of Euler-Poisson system ]]<br />
|M.Feldman<br />
|-<br />
|-<br />
|November 10 <br />
|Philip Isett (MIT)<br />
|[[#Philip Isett| Hölder Continuous Euler Flows ]]<br />
|C.Kim<br />
|-<br />
|-<br />
|November 17<br />
|Lei Wu<br />
|[[#Lei Wu | Geometric Correction for Diffusive Expansion in Neutron Transport Equation ]]<br />
|C.Kim<br />
|-<br />
|-<br />
|December 1<br />
|[http://orion.math.iastate.edu/xhnguyen Xuan Hien Nguyen] (Iowa State University)<br />
|[[#Xuan Hien Nguyen| Gluing constructions for self-similar surfaces under mean curvature flow ]]<br />
|Angenent<br />
|-<br />
|}<br />
<br />
<br />
<br />
<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 14 at 4pm in B139 (TUESDAY), joint with Analysis<br />
|[http://www.math.univ-toulouse.fr/~roque/ Jean-Michel Roquejoffre (Toulouse)]<br />
|[[#Jean-Michel Roquejoffre (Toulouse) |<br />
Front propagation in the presence of integral diffusion. ]]<br />
|Zlatos<br />
|-<br />
|-<br />
|February 10<br />
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]<br />
|[[#Myoungjean Bae (POSTECH) |<br />
Free Boundary Problem related to Euler-Poisson system. ]]<br />
|Feldman<br />
|-<br />
|-<br />
|February 24<br />
|[http://www2.math.umd.edu/~jefftan/ Changhui Tan (Maryland)]<br />
|[[#Changhui Tan (Maryland) |<br />
Global classical solution and long time behavior of macroscopic flocking models. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
Parabolic equations in time-varying domains. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 10<br />
|[http://math.uchicago.edu/~jiahao/ Hao Jia (University of Chicago)]<br />
|[[#Hao Jia (University of Chicago) |<br />
Long time dynamics of energy critical defocusing wave equation with<br />
radial potential in 3+1 dimensions. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 31<br />
|[http://www.mth.kcl.ac.uk/staff/a_pushnitski.html Alexander Pushnitski (King's College London)]<br />
|[[#Alexander Pushnitski (King's College) |<br />
An inverse spectral problem for Hankel operators. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 21<br />
|[http://people.math.gatech.edu/~panrh/ Ronghua Pan (Georgia Tech)]<br />
|[[#Ronghua Pan (Georgia Tech) |<br />
Compressible Navier-Stokes-Fourier system with temperature dependent dissipation. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 9<br />
|Greg Drugan (U. of Washington)<br />
|[[#Greg Drugan (U. of Washington) |<br />
Construction of immersed self-shrinkers]]<br />
|Angenent<br />
|-<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 25<br />
|Myeongju Chae (Hankyong National University visiting UW)<br />
|[[#Myeongju Chae (Hankyong National University) |<br />
On the global classical solution of the Keller-Segel-Navier -Stokes system and its asymptotic behavior. ]]<br />
|Kiselev<br />
|-<br />
myeongju Chae <br />
|-<br />
|December 2<br />
|Xiaojie Wang<br />
|[[#Xiaojie Wang (Stony Brook University) |<br />
Uniqueness of Ricci flow solutions on noncompact manifolds. ]]<br />
|Wang<br />
|-<br />
|-<br />
|December 16<br />
|Antonio Ache(Princeton)<br />
|[[#Antonio Ache(Princeton) |<br />
Ricci Curvature and the manifold learning problem. NOTE: Due to final exams, this seminar will be held in B231. ]]<br />
|Viaclovsky<br />
|-<br />
|}<br />
<br />
<br />
<br />
= Seminar Schedule Spring 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|-<br />
|February 4<br />
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]<br />
|[[#Myoungjean Bae (POSTECH) |<br />
Transonic shocks for Euler-Poisson system and related problems]]<br />
|Feldman<br />
|-<br />
|-<br />
|February 18<br />
|Mike Cullen (Met. Office, UK)<br />
|[[ #Mike Cullen (Met. Office, UK)|<br />
Modelling the uncertainty in predicting large-scale atmospheric circulations.]]<br />
|Feldman<br />
|-<br />
|-<br />
|March 18<br />
|Mohammad Ghomi(Math. Georgia Tech)<br />
|[[ #Mohammad Ghomi(Math. Georgia Tech)|<br />
Tangent lines, inflections, and vertices of closed curves.]]<br />
|Angenent<br />
|-<br />
|-<br />
|April 8<br />
|[http://www.maths.ox.ac.uk/contact/details/xiang Wei Xiang (Oxford)]<br />
|[[ #Wei Xiang (Oxford)|<br />
Shock Diffraction Problem to the<br />
Two Dimensional Nonlinear Wave System and Potential Flow Equation.]]<br />
|Feldman<br />
|-<br />
|-<br />
|Thursday April 25, Time: 3:30, Room: 901 Van Vleck (Note: Special Day, Room)<br />
|[http://math.wvu.edu/~adriant/CV1.html Adrian Tudorascu (West Virginia University)]<br />
|[[ #Adrian Tudorascu (West Virginia University)|One-dimensional pressureless <br />
Euler/Euler-Poisson systems with/without viscosity<br />
.]]<br />
|Feldman<br />
|-<br />
|-<br />
|May 6<br />
|[http://www.icmat.es/dcordoba Diego Cordoba (Madrid)]<br />
|[[#Diego Cordoba (Madrid) |<br />
Interface dynamics for incompressible fluids. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Fall 2012 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 17<br />
|[http://www.math.wisc.edu/~bwang/ Bing Wang (UW Madison)]<br />
|[[#Bing Wang (UW Madison)|<br />
On the regularity of limit space]]<br />
|local<br />
|-<br />
|-<br />
|October 15<br />
|[http://www.math.umn.edu/~polacik/ Peter Polacik (University of Minnesota)]<br />
|[[#Peter Polacik (University of Minnesota)|<br />
Exponential separation between positive and sign-changing solutions and its applications]]<br />
|Zlatos<br />
|-<br />
|-<br />
|November 26<br />
|[http://www.math.wisc.edu/~kchoi/ Kyudong Choi (UW Madison)]<br />
|[[#Kyudong Choi (UW Madison)|<br />
Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations]]<br />
|local<br />
|-<br />
|-<br />
|December 10<br />
|[http://www.math.wisc.edu/~yaoyao/ Yao Yao (UW Madison)]<br />
|[[#Yao Yao (UW Madison)|<br />
Confinement for nonlocal interaction equation with repulsive-attractive kernels]]<br />
|local<br />
|-<br />
|}<br />
<br />
<br />
= Seminar Schedule Spring 2012 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Feb 6<br />
|Yao Yao (UCLA)<br />
|[[#Yao Yao (UCLA)|<br />
Degenerate diffusion with nonlocal aggregation: behavior of solutions]]<br />
|Kiselev<br />
|-<br />
|March 12<br />
| Xuan Hien Nguyen (Iowa State)<br />
|[[#Xuan Hien Nguyen (Iowa State)|<br />
Gluing constructions for solitons and self-shrinkers under mean curvature flow]]<br />
|Angenent<br />
|-<br />
|March 21(Wednesday!), Room 901 Van Vleck<br />
|Nestor Guillen (UCLA)<br />
|[[#Nestor Guillen (UCLA)|<br />
The local geometry of maps with c-convex potentials]]<br />
|Feldman<br />
|-<br />
|March 26<br />
|Vlad Vicol (University of Chicago)<br />
|[[#Vlad Vicol (U Chicago)|<br />
Shape dependent maximum principles and applications]]<br />
|Kiselev<br />
|-<br />
|April 9<br />
|Charles Smart (MIT) <br />
|[[#Charles Smart (MIT)|<br />
PDE methods for the Abelian sandpile<br />
]]<br />
|Seeger<br />
|-<br />
|April 16<br />
|Jiahong Wu (Oklahoma)<br />
|[[#Jiahong Wu (Oklahoma State)|<br />
The 2D Boussinesq equations with partial dissipation]]<br />
|Kiselev<br />
|-<br />
|April 23<br />
|Joana Oliveira dos Santos Amorim (Universite Paris Dauphine)<br />
|[[#Joana Oliveira dos Santos Amorim (Universite Paris Dauphine)|<br />
A geometric look on Aubry-Mather theory and a theorem of Birkhoff]]<br />
|Bolotin<br />
|-<br />
|April 27 (Colloquium. Friday at 4pm, in Van Vleck B239)<br />
|Gui-Qiang Chen (Oxford) <br />
|[[#Gui-Qiang Chen (Oxford) |<br />
Nonlinear Partial Differential Equations of Mixed Type ]]<br />
|Feldman<br />
|-<br />
|May 14<br />
|Jacob Glenn-Levin (UT Austin)<br />
|[[#Jacob Glenn-Levin (UT Austin)|<br />
Incompressible Boussinesq equations in borderline Besov spaces]]<br />
|Kiselev<br />
|}<br />
<br />
<br />
= Seminar Schedule Fall 2011 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Oct 3<br />
|Takis Souganidis (Chicago)<br />
|[[#Takis Souganidis (Chicago)|<br />
''Stochastic homogenization of the G-equation'']]<br />
|Armstrong<br />
<br />
|-<br />
|Oct 10<br />
|Scott Armstrong (UW-Madison)<br />
|[[#Scott Armstrong (UW-Madison)|<br />
''Partial regularity for fully nonlinear elliptic equations'']]<br />
|Local speaker<br />
<br />
|-<br />
|Oct 17<br />
|Russell Schwab (Carnegie Mellon)<br />
|[[#Russell Schwab (Carnegie Mellon)|<br />
''On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)'']]<br />
|Armstrong<br />
|-<br />
|October 24 ( with Geometry/Topology seminar)<br />
|[http://math.univ-lyon1.fr/~ovsienko/ Valentin Ovsienko] (University of Lyon)<br />
|[[#Valentin Ovsienko (University of Lyon)|<br />
''The pentagram map and generalized friezes of Coxeter'']]<br />
|Marí Beffa<br />
|-<br />
|-<br />
|Oct 31<br />
|Adrian Tudorascu (West Virginia University)<br />
|[[#Adrian Tudorascu (West Virginia University)|<br />
''Weak Lagrangian solutions for the Semi-Geostrophic system in physical space'']]<br />
|Feldman<br />
|-<br />
|Nov 7<br />
|James Nolen (Duke)<br />
|[[#James Nolen (Duke)|<br />
''Normal approximation for a random elliptic PDE'']]<br />
|Armstrong<br />
|-|-<br />
|Nov 21 (Joint with [http://www.math.wisc.edu/~seeger/curr.html Analysis] seminar)<br />
|Betsy Stovall (UCLA)<br />
|[[#Betsy Stovall (UCLA)| <br />
''Scattering for the cubic Klein--Gordon equation in two dimensions'']]<br />
|Seeger<br />
|-<br />
|Dec 5<br />
|Charles Smart (MIT)<br />
|[[#Charles Smart (MIT)|<br />
''Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian'']]<br />
|Armstrong<br />
|}<br />
<br />
= Seminar Schedule Spring 2011 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Jan 24<br />
|Bing Wang (Princeton)<br />
|[[#Bing Wang (Princeton)|<br />
''The Kaehler Ricci flow on Fano manifold '']]<br />
|Viaclovsky<br />
|-<br />
|Mar 15 (TUESDAY) at 4pm in B139 (joint wit Analysis)<br />
|Francois Hamel (Marseille)<br />
|[[#Francois Hamel (Marseille)|<br />
''Optimization of eigenvalues of non-symmetric elliptic operators'']]<br />
|Zlatos<br />
|-<br />
|Mar 28<br />
|Juraj Foldes (Vanderbilt)<br />
|[[#Juraj Foldes (Vanderbilt)|<br />
''Symmetry properties of parabolic problems and their applications'']]<br />
|Zlatos<br />
|-<br />
|Apr 11<br />
|Alexey Cheskidov (UIC)<br />
|[[#Alexey Cheskidov (UIC)|<br />
''Navier-Stokes and Euler equations: a unified approach to the problem of blow-up'']]<br />
|Kiselev<br />
|-<br />
|Date TBA<br />
|Mikhail Feldman (UW Madison)<br />
|''TBA''<br />
|Local speaker<br />
|-<br />
|Date TBA<br />
|Sigurd Angenent (UW Madison)<br />
|''TBA''<br />
|Local speaker<br />
|-<br />
|}<br />
= Seminar Schedule Fall 2010 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 13<br />
|Fausto Ferrari (Bologna)<br />
|[[#Fausto Ferrari (Bologna)|<br />
''Semilinear PDEs and some symmetry properties of stable solutions'']]<br />
|Feldman<br />
|-<br />
|Sept 27<br />
|Arshak Petrosyan (Purdue)<br />
|[[#Arshak Petrosyan (Purdue)|<br />
''Nonuniqueness in a free boundary problem from combustion'']]<br />
|Feldman<br />
|-<br />
|Oct 7, Thursday, 4:30 pm, Room: 901 Van Vleck. '''Special day, time & room.'''<br />
|Changyou Wang (U. of Kentucky)<br />
|[[#Changyou Wang (U. of Kentucky)|<br />
''Phase transition for higher dimensional wells'']]<br />
|Feldman<br />
|-<br />
|Oct 11<br />
|Philippe LeFloch (Paris VI)<br />
|[[#Philippe LeFloch (Paris VI)|<br />
''Kinetic relations for undercompressive shock waves and propagating phase boundaries'']]<br />
|Feldman<br />
|-<br />
|Oct 29 Friday 2:30pm, Room: B115 Van Vleck. '''Special day, time & room.'''<br />
|[http://www.ima.umn.edu/~imitrea/ Irina Mitrea] (IMA)<br />
|[[#Irina Mitrea |<br />
''Boundary Value Problems for Higher Order Differential Operators'']]<br />
|[https://www.math.wisc.edu/~wimaw/ WiMaW]<br />
|-<br />
|-<br />
|Nov 1<br />
|Panagiota Daskalopoulos (Columbia U)<br />
|[[#Panagiota Daskalopoulos (Columbia U)|<br />
''Ancient solutions to geometric flows'']]<br />
|Feldman<br />
|-<br />
|Nov 8<br />
|Maria Gualdani (UT Austin)<br />
|[[#Maria Gualdani (UT Austin)|<br />
''A nonlinear diffusion model in mean-field games'']]<br />
|Feldman<br />
|-<br />
|Nov 18 Thursday 1:20pm Room: 901 Van Vleck '''Special day & time.'''<br />
|Hiroshi Matano (Tokyo University) <br />
|[[#Hiroshi Matano (Tokyo University)|<br />
''Traveling waves in a sawtoothed cylinder and their homogenization limit'']]<br />
|Angenent & Rabinowitz<br />
|-<br />
|Nov 29<br />
|Ian Tice (Brown University)<br />
|[[#Ian Tice (Brown University)|<br />
''Global well-posedness and decay for the viscous surface wave<br />
problem without surface tension'']]<br />
|Feldman<br />
|-<br />
|Dec. 8 Wed 2:25pm, Room: 901 Van Vleck. '''Special day, time & room.'''<br />
|Hoai Minh Nguyen (NYU-Courant Institute)<br />
|[[#Hoai Minh Nguyen (NYU-Courant Institute)|<br />
''Cloaking via change of variables for the Helmholtz equation'']]<br />
|Feldman<br />
|-<br />
|}<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
==Abstracts==<br />
<br />
===Dan Knopf===<br />
<br />
Title: Non-K&auml;hler Ricci flow singularities that converge to K&auml;hler-Ricci solitons<br />
<br />
Abstract: We describe Riemannian (non-K&auml;hler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking K&auml;hler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered by Feldman, Ilmanen, and Knopf in 2003. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-K&auml;hler solutions of Ricci flow that become asymptotically K&auml;hler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of K&auml;hler metrics under Ricci flow.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Sam Krupa===<br />
<br />
Title: Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case<br />
<br />
Abstract: For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (Panov). This single entropy result was proven again by De Lellis, Otto and Westdickenberg in 2004. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In our new work, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case. This is joint work with A. Vasseur.<br />
<br />
<br />
===Maja Taskovic===<br />
<br />
Title: Exponential tails for the non-cutoff Boltzmann equation<br />
<br />
Abstract: The Boltzmann equation models the motion of a rarefied gas, in which particles interact through binary collisions, by describing the evolution of the particle density function. The effect of collisions on the density function is modeled by a bilinear integral operator (collision operator) which in many cases has a non-integrable angular kernel. For a long time the equation was simplified by assuming that this kernel is integrable (the so called Grad's cutoff) with a belief that such an assumption does not affect the equation significantly. However, in the last 20 years it has been observed that a non-integrable singularity carries regularizing properties which motivates further analysis of the equation in this setting.<br />
<br />
We study behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime by examining the generation and propagation of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. For this purpose we introduce Mittag-Leffler moments which can be understood as a generalization of exponential moments. An interesting aspect of this result is that the singularity rate of the angular kernel affects the order of tails that can be shown to propagate in time. This is based on joint works with Alonso, Gamba, Pavlovic and Gamba, Pavlovic.<br />
<br />
<br />
===Ashish Kumar Pandey===<br />
<br />
Title: Instabilities in shallow water wave models<br />
<br />
Abstract: Slow modulations in wave characteristics of a nonlinear, periodic traveling wave in a dispersive medium may develop non-trivial structures which evolve as it propagates. This phenomenon is called modulational instability. In the context of water waves, this phenomenon was observed by Benjamin and Feir and, independently, by Whitham in Stokes' waves. I will discuss a general mechanism to study modulational instability of periodic traveling waves which can be applied to several classes of nonlinear dispersive equations including KdV, BBM, and regularized Boussinesq type equations.<br />
<br />
<br />
===Khai Nguyen===<br />
<br />
Title: Burgers Equation with Some Nonlocal Sources<br />
<br />
Abstract: Consider the Burgers equation with some nonlocal sources, which were derived from models of nonlinear wave with constant frequency. This talk will present some recent results on the global existence of entropy weak solutions, priori estimates, and a uniqueness result for both Burgers-Poisson and Burgers-Hilbert equations. Some open questions will be discussed.<br />
<br />
===Hongwei Gao=== <br />
<br />
Title: Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations<br />
<br />
Abstract: In this talk, we discuss the stochastic homogenization of certain nonconvex Hamilton-Jacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of (level-set) convex Hamiltonians and a sequence of (level-set) concave Hamiltonians through the min-max formula. We provide a monotonicity assumption on the contact values between those stably paired Hamiltonians so as to guarantee the stochastic homogenization. If time permits, we will talk about some homogenization results when the monotonicity assumption breaks down.<br />
<br />
===Huy Nguyen===<br />
<br />
Title : Compressible fluids and active potentials<br />
<br />
Abstract: We consider a class of one dimensional compressible systems with degenerate diffusion coefficients. We establish the fact that the solutions remain smooth as long as the diffusion coefficients do not vanish, and give local and global existence results. The models include the barotropic compressible Navier-Stokes equations, shallow water systems and the lubrication approximation of slender jets. In all these models the momentum equation is forced by the gradient of a solution-dependent potential: the active potential. The method of proof uses the Bresch-Desjardins entropy and the analysis of the evolution of the active potential.<br />
<br />
===In-Jee Jeong===<br />
<br />
Title: Singularity formation for the 3D axisymmetric Euler equations<br />
<br />
Abstract: We consider the 3D axisymmetric Euler equations on exterior domains $\{ (x,y,z) : (1 + \epsilon|z|)^2 \le x^2 + y^2 \} $ for any $\epsilon > 0$ so that we can get arbitrarily close to the exterior of a cylinder. We construct a strong local well-posedness class, and show that within this class there exist compactly supported initial data which blows up in finite time. The local well-posedness class consists of velocities which are uniformly Lipschitz in space and have finite energy. Our results were inspired by recent works of Hou-Luo, Kiselev-Sverak, and many others, and the proof builds up on our previous works on 2D Euler and Boussinesq systems. This is joint work with Tarek Elgindi.<br />
<br />
===Jeff Calder===<br />
<br />
Title: Nonlinear PDE continuum limits in data science and machine learning<br />
<br />
Abstract: We will present some recent results on PDE continuum limits for (random) discrete problems in data science and machine learning. All of the problems satisfy a type of discrete comparison/maximum principle and so the continuum PDEs are properly interpreted in the viscosity sense. We will present results for nondominated sorting, convex hull peeling, and graph-based semi-supervised learning. Nondominated sorting is an algorithm for arranging points in Euclidean space into layers by repeatedly peeling away coordinatewise minimal points, and the continuum PDE turns out to be a Hamilton-Jacobi equation. Convex hull peeling is used to order data by repeatedly peeling the vertices of the convex hull, and the continuum limit is motion by a power of Gauss curvature. Finally, a recently proposed class of graph-based learning problems have PDE continuum limits corresponding to weighted p-Laplace equations. In each case the continuum PDEs provide insights into the data science/engineering problems, and suggest avenues for fast approximate algorithms based on the PDE interpretations.<br />
<br />
===Hitoshi Ishii===<br />
<br />
Title: Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory<br />
<br />
Abstract: In the lecture, I discuss two asymptotic problems related to Hamilton-Jacobi equations. One concerns the long-time behavior of solutions of time evolutionary Hamilton-Jacobi equations and the other is the so-called vanishing discount problem for stationary Hamilton-Jacobi equations. The last two decades have seen a fundamental importance of weak KAM theory in the asymptotic analysis of Hamilton-Jacobi equations. I explain briefly the Aubry sets and Mather measures from weak KAM theory and their use in the analysis of the two asymptotic problems above.<br />
<br />
===Mihaela Ifrim===<br />
<br />
Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation<br />
<br />
Our goal is to take a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation. While this problem is known to be both completely integrable and globally well-posed in $L^2$, much less seems to be known concerning its long time dynamics. We present that for small localized data the solutions have (nearly) dispersive dynamics almost globally in time. An additional objective is to revisit the $L^2$ theory for the Benjamin-Ono equation and provide a simpler, self-contained approach. This is joined work with Daniel Tataru.<br />
<br />
===Longjie Zhang===<br />
<br />
On curvature flow with driving force starting as singular initial curve in the plane<br />
<br />
We consider a family of axisymmetric curves evolving by its mean curvature with driving force in the plane. However, the initial curve is oriented singularly at origin. We investigate this problem by level set method and give some criteria to judge whether the interface evolution is fattening or not. In the end, we can classify the solutions into three categories and provide the asymptotic behavior in each category. Our main tools in this paper are level set method and intersection number principle.<br />
<br />
===Jaeyoung Byeon===<br />
<br />
Title: Patterns formation for elliptic systems with large interaction forces<br />
<br />
Abstract: Nonlinear elliptic systems arising from nonlinear Schroedinger systems have simple looking reaction terms. The corresponding energy for the reaction terms can be expressed as quadratic forms in terms of density functions. The i, j-th entry of the matrix for the quadratic form represents the interaction force between the components i and j of the system. If the sign of an entry is positive, the force between the two components is attractive; on the other hand, if it is negative, it is repulsive. When the interaction forces between different components are large, the network structure of attraction and repulsion between components might produce several interesting patterns for solutions. As a starting point to study the general pattern formation structure for systems with a large number of components, I will first discuss the simple case of 2-component systems, and then the much more complex case of 3-component systems.<br />
<br />
<br />
===Tuoc Phan===<br />
Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application.<br />
<br />
Abstract: In this talk, we first introduce a problem on the existence of global time smooth solutions for a system of cross-diffusion equations. We then recall some classical results on regularity theories, and show that to solve our problem, new results on regularity theory estimates of Calderon-Zygmund type for gradients of solutions to a class of parabolic equations in Lebesgue spaces are required. We then discuss a result on Calderon-Zygmnud type estimate in the concrete setting to solve our<br />
mentioned problem regarding the system of cross-diffusion equations. The remaining part of the talk will be focused on some new generalized results on regularity gradient estimates for some general class of quasi-linear parabolic equations. Regularity estimates for gradients of solutions in Lorentz spaces will be presented. Ideas of the proofs for the results are given.<br />
<br />
===Hiroyoshi Mitake===<br />
Derivation of multi-layered interface system and its application<br />
<br />
Abstract: In this talk, I will propose a multi-layered interface system which can be formally derived by the singular limit of the weakly coupled system of the Allen-Cahn equation. By using the level set approach, this system can be written as a quasi-monotone degenerate parabolic system. We give results of the well-posedness of viscosity solutions, and study the singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.<br />
<br />
<br />
===Dongnam Ko===<br />
On the emergence of local flocking phenomena in Cucker-Smale ensembles<br />
<br />
Emergence of flocking groups are often observed in many complex network systems. The Cucker-Smale model is one of the flocking model, which describes the dynamics of attracting particles. This talk concerns time-asymptotic behaviors of Cucker-Smale particle ensembles, especially for mono-cluster and bi-cluster flockings. The emergence of flocking phenomena is determined by sufficient initial conditions, coupling strength, and communication weight decay. Our asymptotic analysis uses the Lyapunov functional approach and a Lagrangian formulation of the coupled system. We derive a system of differential inequalities for the functionals that measure the local fluctuations and group separations along particle trajectories. The bootstrapping argument is the key idea to prove the gathering and separating behaviors of Cucker-Smale particles simultaneously.<br />
<br />
===Sameer Iyer===<br />
Title: Global-in-x Steady Prandtl Expansion over a Moving Boundary.<br />
<br />
Abstract: I will outline the proof that steady, incompressible Navier-Stokes flows posed over the moving boundary, y = 0, can be decomposed into Euler and Prandtl flows globally in the tangential variable, assuming a sufficiently small velocity mismatch. The main obstacles in the analysis center around obtaining sharp decay rates for the linearized profiles and the remainders. The remainders are controlled via a high-order energy method, supplemented with appropriate embedding theorems, which I will present.<br />
<br />
===Jingrui Cheng===<br />
<br />
A 1-D semigeostrophic model with moist convection.<br />
<br />
We consider a simplified 1-D model of semigeostrophic system with moisture, which describes moist convection in a single column in the atmosphere. In general, the solution is non-continuous and it is nontrivial part of the problem to find a suitable definition of weak solutions. We propose a plausible definition of such weak solutions which describes the evolution of the probability distribution of the physical quantities, so that the equations hold in the sense of almost everywhere. Such solutions are constructed from a discrete scheme which obeys the physical principles. This is joint work with Mike Cullen, together with Bin Cheng, John Norbury and Matthew Turner.<br />
<br />
===Donghyun Lee===<br />
<br />
We construct a unique global-in-time solution to the Vlasov-Poisson-Boltzmann system in convex domains with the diffuse boundary condition. Moreover we prove an exponential convergence of distribution function toward the global Maxwellian.<br />
<br />
===Myoungjean Bae===<br />
<br />
3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system.<br />
<br />
I will present a recent result on the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler–Poisson system in a cylinder supplemented with non-small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl (=angular momentum density) component. This talk is based on a joint work with S. Weng (Wuhan University, China).<br />
<br />
===Jingchen Hu===<br />
<br />
Shock Reflection and Diffraction Problem with Potential Flow Equation<br />
<br />
In this talk, we will present our work on nonsymmetric shock reflection and diffraction problem, the equation concerned is potential flow equation, which is a simplification of Euler System, mainly based on the assumption that flow has no vortex. We showed in both nonsymmetric reflection case and diffraction case, that physically admissible solution does not exist. This implies that the formation of vortex is essential to maintain the structural stability of shock reflection and diffraction.<br />
<br />
===Xiaoqin Guo===<br />
<br />
Quantitative homogenization and Harnack inequality for a degenerate discrete nondivergence form random operator.<br />
<br />
In the d-dimensional integer lattice $\mathbb Z^d$, $d\ge 2$, we consider a discrete non-divergence form difference operator<br />
$$<br />
L_a u(x)=\sum_{i=1}^d a_i(x)[u(x+e_i)+u(x-e_i)-2u(x)]<br />
$$<br />
where $a(x)=diag(a_1(x),..., a_d(x)), x\in\mathbb Z^d$ are random nonnegative diagonal matrices which are identically distributed and independent and with a positive expectation.<br />
A difficulty in studying this problem is that coefficients are allowed to be zero. In this talk, using random walks in random media and its percolative structure, we will present a Harnack inequality and quantitative homogenization result for this random operator. Joint work with N.Berger, M.Cohen and J.-D. Deuschel.<br />
<br />
===Ru-Yu Lai===<br />
<br />
Inverse problems for Maxwell's equations and its application.<br />
<br />
This talk will illustrate the application of complex geometrical optics (CGO) solutions to Maxwell's equations. <br />
First, I will explain the increasing stability behavior of coefficients for Maxwell equations. <br />
In particular, by using CGO solutions, the stability estimate of the conductivity is improving when frequency is growing. <br />
Second, I will describe the construction of new families of accelerating and almost nondiffracting beams for Maxwell's equations. <br />
They have the form of wave packets that propagate along circular trajectories while almost preserving a trasverse intensity profile.<br />
<br />
===Norbert Pozar===<br />
<br />
<br />
Viscosity solutions for the crystalline mean curvature flow<br />
<br />
In this talk I will present some recent results concerning the analysis<br />
of the level set formulation of the crystalline mean curvature flow.<br />
The crystalline mean curvature, understood as the first variation of an<br />
anisotropic surface energy with an anisotropy whose Wulff shape is a<br />
polytope, is a singular quantity, nonlocal on the flat parts of the<br />
crystal surface. Therefore the level set equation is not a usual PDE and<br />
does not fit into the classical viscosity solution framework. Its<br />
well-posedness in dimensions higher than two was an open problem until<br />
very recently. In a joint work with Yoshikazu Giga (U. of Tokyo), we<br />
introduce a new notion of viscosity solutions for this problem and<br />
establish its well-posedness for compact crystals in an arbitrary<br />
dimension, including a comparison principle and the stability with<br />
respect to approximation by a smooth anisotropic mean curvature flow.<br />
<br />
<br />
<br />
===Sigurd Angenent===<br />
The Huisken-Hamilton-Gage theorem on compact convex solutions to MCF shows that in forward time all solutions do the same thing, namely, they shrink to a point and become round as they do so. Even though MCF is ill-posed in backward time there do exist solutions that are defined for all t<0 , and one can try to classify all such &ldquo;Ancient Solutions.&rdquo; In doing so one finds that there is interesting dynamics associated to ancient solutions. I will discuss what is currently known about these solutions. Some of the talk is based on joint work with Sesum and Daskalopoulos.<br />
<br />
===Serguei Denissov===<br />
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.<br />
<br />
<br />
===Bing Wang===<br />
We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R3. This is a joint work with H.Z. Li.<br />
<br />
===Eric Baer===<br />
We discuss a recent result showing that a characterization of isoperimetric sets (that is, sets minimizing a relative perimeter functional with respect to a fixed volume constraint) inside convex cones as sections of balls centered at the origin (originally due to P.L. Lions and F. Pacella) remains valid for a class of "almost-convex" cones. Key tools include compactness arguments and the use of classically known sharp characterizations of lower bounds for the first nonzero Neumann eigenvalue associated to (geodesically) convex domains in the hemisphere. The work we describe is joint with A. Figalli.<br />
<br />
===Ben Seeger===<br />
I present a homogenization result for pathwise Hamilton-Jacobi equations with "rough" multiplicative driving signals. In doing so, I derive a new well-posedness result when the Hamiltonian is smooth, convex, and positively homogenous. I also demonstrate that equations involving multiple driving signals may homogenize or exhibit blow-up.<br />
<br />
===Sona Akopian===<br />
Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.<br />
<br />
We solve the Cauchy problem associated to an epsilon-parameter family of homogeneous Boltzmann equations for very soft and Coulomb potentials. Proposed in 2013 by Bobylev and Potapenko, the collision kernel that we use is a Dirac mass concentrated at very small angles and relative speeds. The main advantage of such a kernel is that it does not separate its variables (relative speed $u$ and scattering angle $\theta$) and can be viewed as a pseudo-Maxwell molecule collision kernel, which allows for the splitting of the Boltzmann collision operator into its gain and loss terms. Global estimates on the gain term gives us an existence theory for $L^1_k \capL^p$ with any $k\geq 2$ and $p\geq 1.$ Furthermore the bounds we obtain are independent of the epsilon parameter, which allows for analysis of the solutions in the grazing collisions limit, i.e., when epsilon approaches zero and the Boltzmann equation becomes the Landau equation. <br />
<br />
===Sylvia Serfaty===<br />
Mean-Field Limits for Ginzburg-Landau vortices<br />
<br />
Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.<br />
<br />
<br />
===Gui-Qiang Chen===<br />
Supersonic Flow onto Solid Wedges, Multidimensional Shock Waves and Free Boundary Problems<br />
<br />
When an upstream steady uniform supersonic flow, governed by the Euler equations,<br />
impinges onto a symmetric straight-sided wedge, there are two possible steady oblique shock<br />
configurations if the wedge angle is less than the detachment angle -- the steady weak shock<br />
with supersonic or subsonic downstream flow (determined by the wedge angle that is less or larger<br />
than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which<br />
satisfy the entropy conditions.<br />
The fundamental issue -- whether one or both of the steady weak and strong shocks are physically<br />
admissible solutions -- has been vigorously debated over the past eight decades.<br />
In this talk, we discuss some of the most recent developments on the stability analysis<br />
of the steady shock solutions in both the steady and dynamic regimes.<br />
The corresponding stability problems can be formulated as free boundary problems<br />
for nonlinear partial differential equations of mixed elliptic-hyperbolic type, whose<br />
solutions are fundamental for multidimensional hyperbolic conservation laws.<br />
Some further developments, open problems, and mathematical challenges in this direction<br />
are also addressed.<br />
<br />
===Zhenfu Wang===<br />
<br />
Title: Mean field limit for stochastic particle systems with singular forces<br />
<br />
Abstract: We consider large systems of particles interacting through rough interaction kernels. We are able to control the relative entropy between the N-particles distribution<br />
and the expected limit which solves the corresponding McKean-Vlasov PDE. This implies the Mean Field limit to the McKean-Vlasov system together with Propagation of Chaos<br />
through the strong convergence of all the marginals. The method works at the level of the Liouville equation and relies on precise combinatorics results.<br />
<br />
===Andrei Tarfulea===<br />
We consider a model for three-dimensional fluid flow on the torus that also keeps track of the local temperature. The momentum equation is the same as for Navier-Stokes, however the kinematic viscosity grows as a function of the local temperature. The temperature is, in turn, fed by the local dissipation of kinetic energy. Intuitively, this leads to a mechanism whereby turbulent regions increase their local viscosity and<br />
dissipate faster. We prove a strong a priori bound (that would fall within the Ladyzhenskaya-Prodi-Serrin criterion for ordinary Navier-Stokes) on the thermally weighted enstrophy for classical solutions to the coupled system.<br />
<br />
===Siao-hao Guo===<br />
Analysis of Velázquez's solution to the mean curvature flow with a type II singularity<br />
<br />
Velázquez discovered a solution to the mean curvature flow which develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution, the rescaled flow converges in the C^0 sense to a minimal hypersurface which is tangent to Simons' cone at infinity. In this talk, we will present that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we will show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form. This is a joint work with N. Sesum.<br />
<br />
===Jianfeng Lu===<br />
Evolution of crystal surfaces: from mesoscopic to continuum models<br />
<br />
In this talk, we will discuss some of our recent results on understanding various models for crystal surface evolution at different physical scales; in particular, we will focus on the connection of mesoscopic and continuum (PDE) models for crystal surface relaxation and also discuss several PDEs arising from different physical scenarios. Many interesting open problems remain to be studied. Based on joint work with Yuan Gao, Jian-Guo Liu, Dio Margetis and Jeremy Marzuola.<br />
<br />
===Chris Henderson===<br />
A local-in-time Harnack inequality and applications to reaction-diffusion equations<br />
<br />
The classical Harnack inequality requires one to look back in time to obtain a uniform lower bound on the solution to a parabolic equation. In this talk, I will introduce a Harnack-type inequality that allows us to remove this restriction at the expense of a slightly weaker bound. I will then discuss applications of this bound to (time permitting) three non-local reaction-diffusion equations arising in biology. In particular, in each case, this inequality allows us to show that solutions to these equations, which do not enjoy a maximum principle, may be compared with solutions to a related local equation, which does enjoy a maximum principle. Precise estimates of the propagation speed follow from this.<br />
<br />
<br />
===Jeffrey Streets===<br />
Generalized Kahler Ricci flow and a generalized Calabi conjecture<br />
<br />
Generalized Kahler geometry is a natural extension of Kahler geometry with roots in mathematical physics, and is a particularly rich instance of Hitchin's program of `generalized geometries.' In this talk I will discuss an extension of Kahler-Ricci flow to this setting. I will formulate a natural Calabi-Yau type conjecture based on Hitchin/Gualtieri's definition of generalized Calabi-Yau equations, then introduce the flow as a tool for resolving this. The main result is a global existence and convergence result for the flow which yields a partial resolution of this conjecture, and which classifies generalized Kahler structures on hyperKahler backgrounds.<br />
<br />
<br />
===Daniel Spirn===<br />
<br />
Dipole Trajectories in Bose-Einstein Condensates<br />
<br />
Bose-Einstein condensates (BEC) are a state of matter in which supercooled atoms condense into the lowest possible quantum state. One interesting important feature of BECs are the presence of vortices that form when the condensate is stirred with lasers. I will discuss the behavior of these vortices, which interact with both the confinement potential and other vortices. I will also discuss a related inverse problem in which the features of the confinement can be extracted by the propagation of vortex dipoles.<br />
<br />
===Donghyun Lee===<br />
<br />
The Boltzmann equation with specular reflection boundary condition in convex domains<br />
<br />
I will present a recent work (https://arxiv.org/abs/1604.04342) with Chanwoo Kim on the global-wellposedness and stability of the Boltzmann equation in general smooth convex domains.<br />
<br />
===Kevin Zumbrun===<br />
<br />
TITLE: A Stable Manifold Theorem for a class of degenerate evolution equations <br />
<br />
ABSTRACT: We establish a Stable Manifold Theorem, with consequent exponential decay to equilibrium, for a class <br />
<br />
of degenerate evolution equations $Au'+u=D(u,u)$ with A bounded, self-adjoint, and one-to-one, but not invertible, and <br />
<br />
$D$ a bounded, symmetric bilinear map. This is related to a number of other scenarios investigated recently for which the <br />
<br />
associated linearized ODE $Au'+u=0$ is ill-posed with respect to the Cauchy problem. The particular case studied here <br />
<br />
pertains to the steady Boltzmann equation, yielding exponential decay of large-amplitude shock and boundary layers.<br />
<br />
<br />
<br />
===Will Feldman===<br />
<br />
Liquid Drops on a Rough Surface<br />
<br />
I will discuss the problem of determining the minimal energy shape of a liquid droplet resting on a rough solid surface. The shape of a liquid drop on a solid is strongly affected by the micro-structure of the surface on which it rests, where the surface inhomogeneity arises through varying chemical composition and surface roughness. I will explain a macroscopic regularity theory for the free boundary which allows to study homogenization, and more delicate properties like the size of the boundary layer induced by the surface roughness. <br />
<br />
The talk is based on joint work with Inwon Kim. A remark for those attending the weekend conference: this talk will attempt to have as little as possible overlap with I. Kim's conference talks. <br />
<br />
===Ryan Hynd===<br />
<br />
Extremal functions for Morrey’s inequality in convex domains<br />
<br />
A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor. We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold. <br />
<br />
===Gung-Min Gie ===<br />
<br />
Boundary layer analysis of some incompressible flows<br />
<br />
The motions of viscous and inviscid fluids are modeled respectively by the Navier-Stokes and Euler equations. Considering the Navier-Stokes equations at vanishing viscosity as a singular perturbation of the Euler equations, one major problem, still essentially open, is to verify if the Navier-Stokes solutions converge as the viscosity tends to zero to the Euler solution in the presence of physical boundary. In this talk, we study the inviscid limit and boundary layers of some simplified Naiver-Stokes equations by either imposing a certain symmetry to the flow or linearizing the model around a stationary Euler flow. For the examples, we systematically use the method of correctors proposed earlier by J. L. Lions and construct an asymptotic expansion as the sum of the Navier-Stokes solution and the corrector. The corrector, which corrects the discrepancies between the boundary values of the viscous and inviscid solutions, is in fact an (approximating) solution of the corresponding Prandtl type equations. The validity of our asymptotic expansions is then confirmed globally in the whole domain by energy estimates on the difference of the viscous solution and the proposed expansion. This is a joint work with J. Kelliher, M. Lopes Filho, A. Mazzucato, and H. Nussenzveig Lopes.<br />
<br />
===Tau Shean Lim===<br />
<br />
Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators<br />
<br />
We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) with ignition media f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the case of classical diffusion (i.e., Lu = Laplacian(u)) and non-local diffusion (Lu = J*u - u). Our work extends these results to general Levy operators. In particular, we show that a strong diffusivity in the underlying process (in the sense that the first moment of X_1 is infinite) prevents formation of fronts, while a weak diffusivity gives rise to a unique (up to translation) front U and speed c>0.<br />
<br />
===Tarek M. ELgindi===<br />
<br />
Propagation of Singularities in Incompressible Fluids<br />
<br />
We will discuss some recent results on the local and global stability of certain singular solutions to the incompressible 2d Euler equation. We will begin by giving a brief overview of the classical and modern results on the 2d Euler equation--particularly related to well-posedness theory in critical spaces. Then we will present a new well-posedness class which allows for merely Lipschitz continuous velocity fields and non-decaying vorticity. This will be based upon some interesting estimates for singular integrals on spaces with L^\infty scaling. After that we will introduce a class of scale invariant solutions to the 2d Euler equation and describe some of their remarkable properties including the existence of pendulum-like quasi periodic solutions and infinite-time cusp formation in vortex patches with corners. This is a joint work with I. Jeong. <br />
<br />
<br />
===Adrian Tudorascu===<br />
<br />
Hamilton-Jacobi equations in the Wasserstein space of probability measures <br />
<br />
In 2008 Gangbo, Nguyen and Tudorascu showed that certain variational solutions of the Euler-Poisson system in 1D can be regarded as optimal paths for the value-function giving the viscosity solution of some (infinite-dimensional) Hamilton-Jacobi equation whose phase-space is the Wasserstein space of Borel probability measures with finite second moment. At around the same time, Lasry, Lions, and others became interested in such Hamilton-Jacobi equations (HJE) in connection with their developing theory of Mean-Field games. A different approach (less intrinsic than ours) to the notion of viscosity solution was preferred, one that made an immediate connection between HJE in the Wasserstein space and HJE in Hilbert spaces (whose theory was well-studied and fairly well-understood). At the heart of the difference between these approaches lies the choice of the sub/supper-differential in the context of the Wasserstein space (i.e. the interpretation of ``cotangent space'' to this ``pseudo-Riemannian'' manifold) . In this talk I will start with a brief introduction to Mean-Field games and Optimal Transport, then I will discuss the challenges we encounter in the analysis of (our intrinsic) viscosity solutions of HJE in the Wasserstein space. Based on joint work with W. Gangbo.<br />
<br />
===Alexis Vasseur===<br />
<br />
Compressible Navier-Stokes equations with degenerate viscosities <br />
<br />
We will discuss recent results on the construction of weak solutions for<br />
3D compressible Navier-Stokes equations with degenerate viscosities.<br />
The method is based on the Bresch and Desjardins entropy. The main<br />
contribution is to derive MV type inequalities for the weak solutions,<br />
even if it is not verified by the first level of approximation. This<br />
provides existence of global solutions in time, for the compressible<br />
Navier-Stokes equations, in three dimensional space, with large initial<br />
data, possibly vanishing on the vacuum.<br />
<br />
===Minh-Binh Tran===<br />
<br />
Quantum kinetic problems<br />
<br />
After the production of the first BECs, there has been an explosion of research on the kinetic theory associated to BECs. Later, Gardinier, Zoller and collaborators derived a Master Quantum Kinetic Equation for BECs and introduced the terminology ”Quantum Kinetic Theory”. In 2012, Reichl and collaborators made a breakthrough in discovering a new collision operator, which had been missing in the previous works.<br />
My talk is devoted to the description of our recent mathematical works on quantum kinetic theory. The talk will be based on my joint works with Alonso, Gamba (existence, uniqueness, propagation of moments), Nguyen (Maxwellian lower bound), Soffer (coupling Schrodinger–kinetic equations), Escobedo (convergence to equilibrium), Craciun (the analog between the global attractor conjecture in chemical reaction network and the convergence to equilibrium of quantum kinetic equations), Reichl (derivation).<br />
<br />
===David Kaspar===<br />
<br />
Kinetics of shock clustering<br />
<br />
Suppose we solve a (deterministic) scalar conservation law<br />
with random initial data. Can we describe the probability law of the<br />
solution as a stochastic process in x for fixed later time t? The<br />
answer is yes, for certain Markov initial data, and the probability<br />
law factorizes as a product of kernels. These kernels are obtained by<br />
solving a mean-field kinetic equation which most closely resembles the<br />
Smoluchowski coagulation equation. We discuss prior and ongoing work<br />
concerning this and related problems.<br />
<br />
===Brian Weber===<br />
<br />
Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds<br />
<br />
Understanding scalar-flat instantons is crucial for knowing how Ka ̈hler manifolds degenerate. It is known that scalar-flat Kahler 4-manifolds with two symmetries give rise to a pair of linear degenerate-elliptic Heston type equations <br />
of the form x(fxx + fyy) + fx = 0, which were originally studied in mathematical finance. Vice- versa, solving these PDE produce scalar-flat Kahler 4-manifolds. These PDE have been studied locally, but here we describe new global results <br />
and their implications, partic- ularly a classification of scalar-flat metrics on K ̈ahler 4-manifolds and applications for the study of constant scalar curvature and extremal Ka ̈hler metrics.<br />
<br />
<br />
<br />
<br />
<br />
===Tianling Jin===<br />
<br />
Holder gradient estimates for parabolic homogeneous p-Laplacian equations<br />
<br />
We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation <br />
u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u),<br />
where 1<p<\infty. This equation arises from tug-of-war like stochastic games with white noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.<br />
<br />
===Russell Schwab===<br />
<br />
Neumann homogenization via integro-differential methods<br />
<br />
In this talk I will describe how one can use integro-differential methods to attack some Neumann homogenization problems-- that is, describing the effective behavior of solutions to equations with highly oscillatory Neumann data. I will focus on the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with non-co-normal oscillatory Neumann conditions. The analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain. This is joint work with Nestor Guillen.<br />
<br />
===Jingrui Cheng===<br />
<br />
Semi-geostrophic system with variable Coriolis parameter.<br />
<br />
The semi-geostrophic system (abbreviated as SG) is a model of large-scale atmospheric/ocean flows. Previous works about the SG system have been restricted to the case of constant Coriolis force, where we write the equation in "dual coordinates" and solve. This method does not apply for variable Coriolis parameter case. We develop a time-stepping procedure to overcome this difficulty and prove local existence and uniqueness of smooth solutions to SG system. This is joint work with Michael Cullen and Mikhail Feldman.<br />
<br />
<br />
===Paul Rabinowitz===<br />
<br />
On A Double Well Potential System<br />
<br />
We will discuss an elliptic system of partial differential equations of the form<br />
\[<br />
-\Delta u + V_u(x,u) = 0,\;\;x \in \Omega = \R \times \mathcal{D}\subset \R^n, \;\;\mathcal{D} \; bounded \subset \R^{n-1}<br />
\]<br />
\[<br />
\frac{\partial u}{\partial \nu} = 0 \;\;on \;\;\partial \Omega,<br />
\]<br />
with $u \in \R^m$,\; $\Omega$ a cylindrical domain in $\R^n$, and $\nu$ the outward pointing normal to $\partial \Omega$. <br />
Here $V$ is a double well potential with $V(x, a^{\pm})=0$ and $V(x,u)>0$ otherwise. When $n=1, \Omega =\R^m$ and \eqref{*} is a Hamiltonian system of ordinary differential equations. <br />
When $m=1$, it is a single PDE that arises as an Allen-Cahn model for phase transitions. We will <br />
discuss the existence of solutions of \eqref{*} that are heteroclinic from $a^{-}$ to $a^{+}$ or homoclinic to $a^{-}$,<br />
i.e. solutions that are of phase transition type.<br />
<br />
This is joint work with Jaeyoung Byeon (KAIST) and Piero Montecchiari (Ancona).<br />
<br />
===Hong Zhang===<br />
<br />
On an elliptic equation arising from composite material<br />
<br />
I will present some recent results on second-order divergence type equations with piecewise constant coefficients. This problem arises in the study of composite materials with closely spaced interface boundaries, and the classical elliptic regularity theory are not applicable. In the 2D case, we show that any weak solution is piecewise smooth without the restriction of the underling domain where the equation is satisfied. This completely answers a question raised by Li and Vogelius (2000) in the 2D case. Joint work with Hongjie Dong.<br />
<br />
===Aaron Yip===<br />
<br />
Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media<br />
<br />
The talk will describe some results on the behavior of solutions of motion by mean curvature in inhomogeneous media. Emphasis will be put on the pinning and de-pinning transition, continuum limit of discrete spin systems and the motion of interface between patterns.<br />
<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Selection problem for fully nonlinear equations<br />
<br />
Recently, there was substantial progress on the selection problem on the ergodic problem for Hamilton-Jacobi equations, which was open during almost 30 years. In the talk, I will first show a result on the convex Hamilton-Jacobi equation, then tell important problems which still remain. Next, I will mainly focus on a recent joint work with H. Ishii (Waseda U.), and H. V. Tran (U. Wisconsin-Madison) which is about the selection problem for fully nonlinear, degenerate elliptic partial differential equations. I will present a new variational approach for this problem.<br />
<br />
===Nestor Guillen===<br />
<br />
Min-max formulas for integro-differential equations and applications<br />
<br />
We show under minimal assumptions that a nonlinear operator satisfying what is known as a "global comparison principle" can be represented by a min-max formula in terms of very special linear operators (Levy operators, which involve drift-diffusion and integro-differential terms). Such type of formulas have been very useful in the theory of second order equations -for instance, by allowing the representation of solutions as value functions for differential games. Applications include results on the structure of Dirichlet-to-Neumann mappings for fully nonlinear second order elliptic equations.<br />
<br />
===Ryan Denlinger===<br />
<br />
The propagation of chaos for a rarefied gas of hard spheres in vacuum<br />
<br />
We are interested in the rigorous mathematical justification of<br />
Boltzmann's equation starting from the deterministic evolution of<br />
many-particle systems. O. E. Lanford was able to derive Boltzmann's<br />
equation for hard spheres, in the Boltzmann-Grad scaling, on a short<br />
time interval. Improvements to the time in Lanford's theorem have so far<br />
either relied on a small data hypothesis, or have been restricted to<br />
linear regimes. We revisit the small data regime, i.e. a sufficiently<br />
dilute gas of hard spheres dispersing into vacuum; this is a regime<br />
where strong bounds are available globally in time. Subject to the<br />
existence of such bounds, we give a rigorous proof for the propagation<br />
of Boltzmann's ``one-sided'' molecular chaos.<br />
<br />
===Misha Feldman===<br />
<br />
Shock reflection, free boundary problems and degenerate elliptic equations.<br />
<br />
Abstract: We will discuss shock reflection problem for compressible gas dynamics, and von Neumann<br />
conjectures on transition between regular and Mach reflections. We will discuss existence of solutions <br />
of regular reflection structure for potential flow equation, and also regularity of solutions, and<br />
properties of the shock curve (free boundary). Our approach is to reduce the shock reflection problem<br />
to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type. Open problems <br />
will also be discussed, including uniqueness.<br />
The talk is based on the joint works with Gui-Qiang Chen, Myoungjean Bae and Wei Xiang.<br />
<br />
===Jessica Lin===<br />
<br />
Optimal Quantitative Error Estimates in Stochastic<br />
Homogenization for Elliptic Equations in Nondivergence Form<br />
<br />
Abstract: I will present optimal quantitative error estimates in the<br />
stochastic homogenization for uniformly elliptic equations in<br />
nondivergence form. From the point of view of probability theory,<br />
stochastic homogenization is equivalent to identifying a quenched<br />
invariance principle for random walks in a balanced random<br />
environment. Under strong independence assumptions on the environment,<br />
the main argument relies on establishing an exponential version of the<br />
Efron-Stein inequality. As an artifact of the optimal error estimates,<br />
we obtain a regularity theory down to microscopic scale, which implies<br />
estimates on the local integrability of the invariant measure<br />
associated to the process. This talk is based on joint work with Scott<br />
Armstrong.<br />
<br />
===Sergey Bolotin===<br />
<br />
Degenerate billiards<br />
<br />
In an ordinary billiard trajectories of a Hamiltonian system<br />
are elastically reflected when colliding with a hypersurface (scatterer).<br />
If the scatterer is a submanifold of codimension more than one, then collisions are rare.<br />
Trajectories with infinite number of collisions form a lower dimensional dynamical system.<br />
Degenerate billiards appear as limits of ordinary billiards and in celestial mechanics.<br />
<br />
===Moon-Jin Kang===<br />
<br />
On contraction of large perturbation of shock waves, and inviscid limit problems<br />
<br />
This talk will start with the relative entropy method to handle the contraction of possibly large perturbations around viscous shock waves of conservation laws. In the case of viscous scalar conservation law in one space dimension, we obtain $L^2$-contraction for any large perturbations of shocks up to a Lipschitz shift depending on time. Such a time-dependent Lipschitz shift should be constructed from dynamics of the perturbation. In the case of multidimensional scalar conservation law, the perturbations of planar shocks are $L^2$-contractive up to a more complicated shift depending on both time and space variable, which solves a parabolic equation with inhomogeneous coefficient and force terms reflecting the perturbation. As a consequence, the $L^2$-contraction property implies the inviscid limit towards inviscid shock waves. At the end, we handle the contraction properties of admissible discontinuities of the hyperbolic system of conservation laws equipped with a strictly convex entropy.<br />
<br />
<br />
<br />
===Hung Tran===<br />
<br />
Some inverse problems in periodic homogenization of Hamilton--Jacobi equations.<br />
<br />
Abstract: We look at the effective Hamiltonian $\overline{H}$ associated with the Hamiltonian $H(p,x)=H(p)+V(x)$ in the periodic homogenization theory. Our central goal is to understand the relation between $V$ and $\overline{H}$. We formulate some inverse problems concerning this relation. Such type of inverse problems are in general very challenging. I will discuss some interesting cases in both convex and nonconvex settings. Joint work with Songting Luo and Yifeng Yu.<br />
<br />
<br />
===Eric Baer===<br />
<br />
Optimal function spaces for continuity of the Hessian determinant as a distribution.<br />
<br />
Abstract: In this talk we describe a new class of optimal continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on $\mathbb{R}^N$, obtained in collaboration with D. Jerison. Inspired by recent work of Brezis and Nguyen on the distributional Jacobian determinant, we show that the action is continuous on the Besov space $B(2-2/N,N)$ of fractional order, and that all continuity results in this scale of Besov spaces are consequences of this result. A key ingredient in the argument is the characterization of $B(2-2/N,N)$ as the space of traces of functions in the Sobolev space $W^{2,N}(\mathbb{R}^{N+2})$ on the subspace $\mathbb{R}^N$ (of codimension 2). The most elaborate part of the analysis is the construction of a counterexample to continuity in $B(2-2/N,p)$ with $p>N$. Tools involved in this step include the choice of suitable ``atoms" having a tensor product structure and Hessian determinant of uniform sign, formation of lacunary series of rescaled atoms, and delicate estimates of terms in the resulting multilinear expressions.<br />
<br />
===Donghyun Lee===<br />
<br />
FLUIDS WITH FREE-SURFACE AND VANISHING VISCOSITY LIMIT.<br />
<br />
Abstract : Free-boundary problems of incompressible fluids have been studied for several decades. In the viscous case, it is basically solved by Stokes regularity. However, the inviscid case problem is generally much harder, because the problem is purely hyperbolic. In this talk, we approach the problem via vanishing viscosity limit, which is a central problem of fluid mechanics. To correct boundary layer behavior, conormal Sobolev space will be introduced. In the spirit of the recent work by N.Masmoudi and F.Rousset (2012, non-surface tension), we will see how to get local regularity of incompressible free-boundary Euler, taking surface tension into account. This is joint work with Tarek Elgindi.<br />
If possible, we also talk about applying the similar technique to the free-boundary MHD(Magnetohydrodynamics). Especially, we will see that strong zero initial boundary condition is still valid for this coupled PDE. For the general boundary condition (for perfect conductor), however, the problem is still open.<br />
<br />
===Hyung-Ju Hwang===<br />
<br />
The Fokker-Planck equation in bounded domains<br />
<br />
abstract: In this talk, we consider the initial-boundary value problem for the Fokker-Planck equation in an interval or in a bounded domain with absorbing boundary conditions. We discuss a theory of well-posedness of classical solutions for the problem as well as the exponential decay in time, hypoellipticity away from the singular set, and the Holder continuity of the solutions up to the singular set. This is a joint work with J. Jang, J. Jung, and J. Velazquez.<br />
<br />
===Minh-Binh Tran===<br />
<br />
Nonlinear approximation theory for kinetic equations<br />
<br />
Abstract: Numerical resolution methods for the Boltzmann equation plays a very important role in the practical a theoretical study of the theory of rarefied gas. The main difficulty in the approximation of the Boltzmann equation is due to the multidimensional structure of the Boltzmann collision operator. The major problem with deterministic numerical methods using to solve Boltzmann equation is that we have to truncate the domain or to impose nonphysical conditions to keep the supports of the solutions in the velocity space uniformly compact. I<br />
n this talk, we will introduce our new way to make the connection between nonlinear approximation theory and kinetic theory. Our nonlinear wavelet approximation is nontruncated and based on an adaptive spectral method associated with a new wavelet filtering technique. The approximation is proved to converge and preserve many properties of the homogeneous Boltzmann equation. The nonlinear approximation solves the equation without having to impose non-physics conditions on the equation.<br />
<br />
===Bob Jensen===<br />
<br />
Crandall-Lions Viscosity Solutions of Uniformly Elliptic PDEs<br />
<br />
Abstract: I will discuss C-L viscosity solutions of uniformly elliptic partial differential equations for operators with only measurable spatial regularity. E.g., $L[u] = \sum a_{i\,j}(x)\,D_{i\,j}u(x)$ where $a_{i\,j}(x)$ is bounded, uniformly elliptic, and measurable in $x$. In general there isn't a meaningful extension of the C-L viscosity solution definition to operators with measurable spatial dependence. But under uniform ellipticity there is a natural extension. Though there isn't a general comparison principle in this context, we will see that the extended definition is robust and uniquely characterizes the ``right" solutions for such problems.<br />
<br />
===Luis Silvestre===<br />
<br />
A priori estimates for integral equations and the Boltzmann equation.<br />
<br />
Abstract: We will review some results on the regularity of general parabolic integro-differential equations. We will see how these results can be applied in order to obtain a priori estimates for the Boltzmann equation (without cutoff) modelling the evolution of particle density in a dilute gas. We derive a bound in L^infinity for the full Boltzmann equation, and Holder continuity estimates in the space homogeneous case.<br />
<br />
===Connor Mooney===<br />
<br />
Counterexamples to Sobolev regularity for degenerate Monge-Ampere equations<br />
<br />
Abstract: W^{2,1} estimates for the Monge-Ampere equation \det D^2u = f in R^n were first obtained by De Philippis and Figalli in the case that f is bounded between positive constants. Motivated by applications to the semigeostrophic equation, we consider the case that f is bounded but allowed to be zero on some set. In this case there are simple counterexamples to W^{2,1} regularity in dimension n \geq 3 that have a Lipschitz singularity. In contrast, if n = 2 a classical theorem of Alexandrov on the propagation of Lipschitz singularities shows that solutions are C^1. We will discuss a counterexample to W^{2,1} regularity in two dimensions whose second derivatives have nontrivial Cantor part, and also a related result on the propagation of Lipschitz / log(Lipschitz) singularities that is optimal by example.<br />
<br />
===Javier Gomez Serrano===<br />
<br />
Existence and regularity of rotating global solutions for active scalars<br />
<br />
A particular kind of weak solutions for a 2D active scalar equation are the so called patches, i.e., solutions for which the scalar is a step function taking one value inside a moving region and another in the complement. The evolution of such distribution is completely determined by the evolution of the boundary, allowing the problem to be treated as a non-local one dimensional equation for the contour. In this talk we will discuss the existence and regularity of uniformly rotating solutions for the vortex patch and generalized surface quasi-geostrophic (gSQG) patch equation. We will also outline the proof for the smooth (non patch) SQG case. Joint work with Angel Castro and Diego Cordoba.<br />
<br />
===Yifeng Yu===<br />
<br />
G-equation in the modeling of flame propagation.<br />
<br />
Abstract: G-equation is a well known model in turbulent combustion. In<br />
this talk, I will present joint works with Jack Xin about how the<br />
effective burning velocity (turbulent flame speed) depends on the strength<br />
of the ambient fluid (e.g. the speed of the wind) under various G-equation<br />
model.<br />
<br />
===Nam Le===<br />
<br />
Global smoothness of the Monge-Ampere eigenfunctions<br />
<br />
Abstract:<br />
In this talk, I will discuss global smoothness of the eigenfunctions of the Monge-Ampere operator on smooth, bounded and uniformly convex domains in all dimensions. A key ingredient in our analysis is boundary Schauder estimates for certain degenerate Monge-Ampere equations. This is joint work with Ovidiu Savin.<br />
<br />
===Qin Li===<br />
<br />
Kinetic-fluid coupling: transition from the Boltzmann to the Euler<br />
<br />
Abstract: Kinetic equations (the Boltzmann, the neutron transport equation etc.) are known to converge to fluid equations (the Euler, the heat equation etc.) in certain regimes, but when kinetic and fluid regime co-exist, how to couple the two systems remains an open problem. The key is to understand the half-space problem that resembles the boundary layer at the interface. In this talk, I will present a unified proof for the well-posedness of a class of half-space equations with general incoming data, propose an efficient spectral solver, and utilize it to couple fluid with kinetics. Moreover, I will present complete error analysis for the proposed spectral solver. Numerical results will be shown to demonstrate the accuracy of the algorithm.<br />
<br />
===Lu Wang===<br />
<br />
Asymptotic Geometry of Self-shrinkers<br />
<br />
Abstract: In this talk, we will discuss some recent progress towards the conjectural asymptotic behaviors of two-dimensional self-shrinkers of mean curvature flow.<br />
<br />
===Christophe Lacave===<br />
<br />
Well-posedness for 2D Euler in non-smooth domains<br />
<br />
The well-posedness of the Euler system has been of course the matter of many works, but a common point in all the previous studies is that the boundary is at least $C^{1,1}$. In a first part, we will establish the existence of global weak solutions of the 2D incompressible Euler equations for a large class of non-smooth open sets. Existence of weak solutions with $L^p$ vorticity is deduced from an approximation argument, that relates to the so-called $\gamma$-convergence of domains. In a second part, we will prove the uniqueness if the open set is the interior or the exterior of a simply connected domain, where the boundary has a finite number of corners. Although the velocity blows up near these corners, we will get a similar theorem to the Yudovich's result. Theses works are in collaboration with David Gerard-Varet, Evelyne Miot and Chao Wang.<br />
<br />
<br />
===Jun Kitagawa (Toronto)===<br />
<br />
Regularity theory for generated Jacobian equations: from optimal transport to geometric optics<br />
<br />
Equations of Monge-Ampere type arise in numerous contexts, and solutions often exhibit very subtle qualitative and quantitative properties; this is owing to the highly nonlinear nature of the equation, and its degeneracy (in the sense of ellipticity). Motivated by an example from geometric optics, I will talk about the class of Generated Jacobian Equations; recently introduced by Trudinger, this class also encompasses, for example, optimal transport, the Minkowski problem, and the classical Monge-Ampere equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field reflector problems (of interest from a physical and practical point of view). This talk is based on joint works with N. Guillen.<br />
<br />
===Jessica Lin (Madison)===<br />
<br />
Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations<br />
<br />
We establish error estimates for the stochastic homogenization of fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media. Based on the approach of Armstrong and Smart in the elliptic setting, we construct a quantity which captures the geometric behavior of solutions to parabolic equations. The error estimates are shown to be of algebraic order. This talk is based on joint work with Charles Smart.<br />
<br />
<br />
===Yaguang Wang (Shanghai Jiao Tong)===<br />
<br />
Stability of Three-dimensional Prandtl Boundary Layers<br />
<br />
In this talk, we shall study the stability of the Prandtl boundary layer<br />
equations in three space variables. First, we obtain a well-posedness<br />
result of the three-dimensional Prandtl equations under some constraint on<br />
its flow structure. It reveals that the classical Burgers equation plays an<br />
important role in determining this type of flow with special structure,<br />
that avoids the appearance of the complicated secondary flow in the<br />
three-dimensional Prandtl boundary layers. Second, we give an instability<br />
criterion for the Prandtl equations in three space variables. Both of<br />
linear and nonlinear stability are considered. This criterion shows that<br />
the monotonic shear flow is linearly stable for the three dimensional<br />
Prandtl equations if and only if the tangential velocity field direction is<br />
invariant with respect to the normal variable, which is an exact complement<br />
to the above well-posedness result for a special flow. This is a joint work<br />
with Chengjie Liu and Tong Yang.<br />
<br />
<br />
===Benoit Pausader (Princeton)===<br />
<br />
Global smooth solutions for the Euler-Maxwell problem for electrons in 2 dimensions<br />
<br />
It is well known that pure compressible fluids tend to develop shocks, even from small perturbation. We study how self consistent electromagnetic fields can stabilize these fluids. In a joint work with A. Ionescu and Y. Deng, we consider a compressible fluid of electrons in 2D, subject to its own electromagnetic field and to a field created by a uniform background of positively charged ions. We show that small smooth and irrotational perturbations of a uniform background at rest lead to solutions that remain globally smooth, in contrast with neutral fluids. This amounts to proving small data global existence for a system of quasilinear Klein-Gordon equations with different speeds.<br />
<br />
<br />
===Haozhao Li (University of Science and Technology of China)===<br />
<br />
Regularity scales and convergence of the Calabi flow<br />
<br />
We define regularity scales to study the behavior of the Calabi flow. <br />
Based on estimates of the regularity scales, we obtain convergence theorems<br />
of the Calabi flow on extremal K\"ahler surfaces, under the assumption of global existence<br />
of the Calabi flow solutions. Our results partially confirm Donaldson’s conjectural picture for<br />
the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra<br />
assumption that the scalar curvature is uniformly bounded.<br />
<br />
<br />
===Jennifer Beichman (UW-Madison)===<br />
<br />
Nonstandard dispersive estimates and linearized water waves<br />
<br />
In this talk, we focus on understanding the relationship between the decay of a solution to the linearized water wave problem and its initial data. We obtain decay bounds for a class of 1D dispersive equations that includes the linearized water wave. These decay bounds display a surprising growth factor, which we show is sharp. A further exploration leads to a result relating singularities of the initial data at the origin in Fourier frequency to the regularity of the solution.<br />
<br />
===Ben Fehrman (University of Chicago)===<br />
<br />
On The Existence of an Invariant Measure for Isotropic Diffusions in Random Environments<br />
<br />
I will discuss the existence of a unique mutually absolutely continuous invariant measure for isotropic diffusions in random environment, of dimension at least three, which are small perturbations of Brownian motion satisfying a finite range dependence. This framework was first considered in the continuous setting by Sznitman and Zeitouni and in the discrete setting by Bricmont and Kupiainen. The results of this talk should be seen as an extension of their work.<br />
<br />
I will furthermore mention applications of this analysis to the stochastic homogenization of the related elliptic and parabolic equations with random oscillatory boundary data and, explain how the existence of an invariant measure can be used to prove a Liouville property for the environment. In the latter case, the methods were motivated by work in the discrete setting by Benjamini, Duminil-Copin, Kozma and Yadin.<br />
<br />
===Vera Hur=== <br />
<br />
Instabilities in nonlinear dispersive waves<br />
<br />
I will speak on the wave breaking and the modulational instability of nonlinear wave trains in dispersive media. I will begin by a gradient blowup proof for the Boussinesq-Whitham equations for water waves. I will then describe a variational approach to determine instability to long wavelength perturbations for a general class of Hamiltonian systems, allowing for nonlocal dispersion. I will discuss KdV type equations with fractional dispersion in depth. Lastly, I will explain an asymptotics approach for Whitham's equation for water waves, qualitatively reproducing the Benjamin-Feir instability of Stokes waves.<br />
<br />
===Sung-Jin Oh=== <br />
<br />
Global well-posedness of the energy critical Maxwell-Klein-Gordon equation<br />
<br />
The massless Maxwell-Klein-Gordon system describes the interaction between an electromagnetic field (Maxwell) and a charged massless scalar field (massless Klein-Gordon, or wave). <br />
In this talk, I will present a recent proof, joint with D. Tataru, of global well-posedness and scattering of this system for arbitrary finite energy data in the (4+1)-dimensional Minkowski space, in which the PDE is energy critical.<br />
<br />
===Yuan Lou=== <br />
<br />
Asymptotic behavior of the smallest eigenvalue of an elliptic operator and its applications to evolution of dispersal<br />
<br />
We investigate the effects of diffusion and drift on the smallest eigenvalue of an elliptic operator with zero Neumann boundary condition. Various asymptotic behaviors of the smallest eigenvalue, as diffusion and drift rates approach zero or infinity, are derived. As an application, these qualitative results yield some insight into the evolution of dispersal in heterogeneous environments.<br />
<br />
===Diego Cordoba===<br />
<br />
Global existence solutions and geometric properties of the SQG sharp front<br />
<br />
A particular kind of weak solutions for a 2D active scalar are the so called sharp fronts, i.e., solutions for which the scalar<br />
is a step function. The evolution of such distribution is completely determined by the evolution of the boundary, allowing the problem <br />
to be treated as a non-local one dimensional equation for the contour. In this setting we will present several analytical results for the surface <br />
quasi-geostrophic equation (SQG): the existence of convex $C^{\infinity}$ global rotating solutions, elliptical shapes are not rotating solutions (as opposed to 2D Euler equations) and the existence of convex solutions that lose their convexity in finite time.<br />
<br />
<br />
<br />
===Greg Kuperberg===<br />
''Cartan-Hadamard and the Little Prince.''<br />
<br />
===Steve Hofmann===<br />
''Quantitative Rectifiability and Elliptic Equations''<br />
<br />
A classical theorem of F. and M. Riesz states that for a simply connected domain in the complex plane with a rectifiable boundary, harmonic measure and arc length measure on the boundary are mutually absolutely continuous. On the other hand, an example of C. Bishop and P. Jones shows that the latter conclusion may fail, in the absence of some sort of connectivity hypothesis. In this talk, we discuss recent developments in an ongoing program to find scale-invariant, higher dimensional versions of the F. and M. Riesz Theorem, as well as converses. In particular, we discuss substitute results that continue to hold in the absence of any connectivity hypothesis.<br />
<br />
===Xiangwen Zhang===<br />
''Alexandrov's Uniqueness Theorem for Convex Surfaces''<br />
<br />
A classical uniqueness theorem of Alexandrov says that: a closed strictly convex twice differentiable surface in R3 is uniquely determined to within a parallel translation when one gives a proper function of the principle curvatures. We will talk about a PDE proof for this thorem, by using the maximal principle and weak uniqueness continuation theorem of Bers-Nirenberg. Moreover, a stability result related to the uniqueness problem will be mentioned. This is a joint work with P. Guan and Z. Wang.<br />
If time permits, we will also briefly introduce the idea of our recent work on Alexandrov’s theorems for codimension two submanifolds in spacetimes.<br />
<br />
===Xuwen Chen===<br />
''The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-body Evolution''<br />
<br />
We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is focusing and matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D N-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to infinity. We prove that the limiting structure of the density matrices counterbalances this diverging coefficient. We establish the convergence of the BBGKY sequence and hence the propagation of chaos for the focusing quantum many-body system. We derive rigorously the 1D focusing cubic NLS as the mean-field limit of this 3D focusing quantum many-body dynamic and obtain the exact 3D to 1D coupling constant.<br />
<br />
===Kyudong Choi===<br />
''Finite time blow up for 1D models for the 3D Axisymmetric Euler Equations the<br />
2D Boussinesq system''<br />
<br />
In connection with the recent proposal for possible singularity formation at the boundary for solutions of the 3d axi-symmetric incompressible Euler's equations / the 2D Boussinesq system (Luo and Hou, 2013), we study models for the dynamics at the boundary and show that they exhibit a finite-time blow-up from smooth data. This is joint work with T. Hou, A. Kiselev, G. Luo, V. Sverak, and Y. Yao.<br />
<br />
===Myoungjean Bae===<br />
''Recent progress on study of Euler-Poisson system''<br />
<br />
In this talk, I will present recent progress on the following subjects:<br />
(1) Smooth transonic flow of Euler-Poisson system;<br />
(2) Transonic shock of Euler-Poisson system.<br />
This talk is based on collaboration with Ben Duan, Chujing Xie and Jingjing Xiao<br />
<br />
===Philip Isett===<br />
"Hölder Continuous Euler Flows"<br />
<br />
Motivated by the theory of hydrodynamic turbulence, L. Onsager conjectured<br />
in 1949 that solutions to the incompressible Euler equations with Holder<br />
regularity less than 1/3 may fail to conserve energy. C. De Lellis and L.<br />
Székelyhidi, Jr. have pioneered an approach to constructing such irregular<br />
flows based on an iteration scheme known as convex integration. This<br />
approach involves correcting “approximate solutions" by adding rapid<br />
oscillations which are designed to reduce the error term in solving the<br />
equation. In this talk, I will discuss an improved convex integration<br />
framework, which yields solutions with Holder regularity 1/5-, as well as<br />
other related results.<br />
<br />
===Lei Wu===<br />
''Geometric Correction for Diffusive Expansion in Neutron Transport Equation''<br />
<br />
We revisit the diffusive limit of a steady neutron transport equation in a 2-D unit disk with one-speed velocity. The traditional method is Hilbert expansions and boundary layer analysis. We will carefully study the classical theory of the construction of boundary layers, and discuss the necessity and specific method to add the geometric correction.<br />
<br />
===Xuan Hien Nguyen===<br />
In the 1990's, Kapouleas and Traizet constructed new examples of minimal surfaces by desingularizing the intersection of existing ones, such as catenoids and planes, with Scherk surfaces. Using the same strategy, one can prove the existence of new self-translating and self-shrinking surfaces under mean curvature flow. In this talk, we will survey the results obtained so far and propose some generalization and simplification of the techniques.<br />
<br />
<br />
<br />
===Greg Drugan (U. of Washington)===<br />
''Construction of immersed self-shrinkers''<br />
<br />
Abstract: We describe a procedure for constructing immersed<br />
self-shrinking solutions to mean curvature flow. <br />
The self-shrinkers we construct have a rotational symmetry, and<br />
the construction involves a detailed study of geodesics in the<br />
upper-half plane with a conformal metric.<br />
This is a joint work with Stephen Kleene.<br />
<br />
===Guo Luo (Caltech)===<br />
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''<br />
<br />
Abstract:<br />
Whether the 3D incompressible Euler equations can develop a singularity in <br />
finite time from smooth initial data is one of the most challenging problems in <br />
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this <br />
long-standing open question from a numerical point of view, by presenting a class of <br />
potentially singular solutions to the Euler equations computed in axisymmetric <br />
geometries. The solutions satisfy a periodic boundary condition along the axial direction <br />
and no-flow boundary condition on the solid wall. The equations are discretized in space <br />
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially <br />
designed adaptive (moving) meshes that are dynamically adjusted to the evolving <br />
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the <br />
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and <br />
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a <br />
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and <br />
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity <br />
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup <br />
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and <br />
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also <br />
suggests that the blowing-up solution develops a self-similar structure near the point of <br />
the singularity, as the singularity time is approached.<br />
<br />
===Xiaojie Wang(Stony Brook)===<br />
''Uniqueness of Ricci flow solutions on noncompact manifolds''<br />
<br />
Abstract:<br />
Ricci flow is an important evolution equation of Riemannian metrics.<br />
Since it was introduced by R. Hamilton in 1982, it has greatly changed the landscape of riemannian geometry. One of the fundamental question about ricci flow is when is its solution to initial value problem unique. On compact manifold, with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when further restrictions are imposed to the solution. In this talk, we will discuss various conditions that guarantee the uniqueness. In particular, we will discuss in details with the following uniqueness result. Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, on $M\times [0,\epsilon]$ for some $\epsilon>0$, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.<br />
<br />
===Roman Shterenberg(UAB)===<br />
''Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems''<br />
<br />
Abstract: We present a review of the results in multidimensional periodic<br />
and almost-periodic spectral problems. We discuss some recent progress and<br />
old/new ideas used in the constructions. The talk is mostly based on the<br />
joint works with Yu. Karpeshina and L. Parnovski.<br />
<br />
===Antonio Ache(Princeton)===<br />
''Ricci Curvature and the manifold learning problem''<br />
<br />
Abstract: In the first half of this talk we will review several notions of coarse or weak<br />
Ricci Curvature on metric measure spaces which include the works of Lott-Villani, Sturm<br />
and Ollivier. The discussion of the notion of coarse Ricci curvature will serve as <br />
motivation for developing a method to estimate the Ricci curvature of a an embedded<br />
submaifold of Euclidean space from a point cloud which has applications to the Manifold<br />
Learning Problem. Our method is based on combining the notion of ``Carre du Champ"<br />
introduced by Bakry-Emery with a result of Belkin and Niyogi which shows that it is<br />
possible to recover the rough laplacian of embedded submanifolds of the Euclidean space<br />
from point clouds. This is joint work with Micah Warren.<br />
<br />
===Jean-Michel Roquejoffre (Toulouse)===<br />
''Front propagation in the presence of integral diffusion''<br />
<br />
Abstract: In many reaction-diffusion equations, where diffusion is<br />
given by a second order elliptic operator, the solutions<br />
will exhibit spatial transitions whose velocity is asymptotically<br />
linear in time. The situation can be different when the diffusion is of the<br />
integral type, the most basic example being the fractional Laplacian:<br />
the velocity can be time-exponential. We will explain why, and<br />
discuss several situations where this type of fast propagation<br />
occurs.<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Free Boundary Problem related to Euler-Poisson system''<br />
<br />
One dimensional analysis of Euler-Poisson system shows that when incoming <br />
supersonic flow is fixed, transonic shock can be represented as a monotone <br />
function of exit pressure. From this observation, we expect well-posedness <br />
of transonic shock problem for Euler-Poisson system when exit pressure is <br />
prescribed in a proper range. In this talk, I will present recent progress <br />
on transonic shock problem for Euler-Poisson system, which is formulated <br />
as a free boundary problem with mixed type PDE system. <br />
This talk is based on collaboration with Ben Duan(POSTECH), Chujing Xie(SJTU) <br />
and Jingjing Xiao(CUHK).<br />
<br />
===Changhui Tan (University of Maryland)===<br />
''Global classical solution and long time behavior of macroscopic flocking models''<br />
<br />
Abstract: Self-organized behaviors are very common in nature and human societies.<br />
One widely discussed example is the flocking phenomenon which describes<br />
animal groups emerging towards the same direction. Several models such<br />
as Cucker-Smale and Motsch-Tadmor are very successful in characterizing<br />
flocking behaviors. In this talk, we will discuss macroscopic representation<br />
of flocking models. These systems can be interpreted as compressible Eulerian <br />
dynamics with nonlocal alignment forcing. We show global existence of classical solutions and long time <br />
flocking behavior of the system, when initial profile satisfies a threshold condition. On the other hand, another set<br />
of initial conditions will lead to a finite time break down of the system. This<br />
is a joint work with Eitan Tadmor.<br />
<br />
===Hongjie Dong (Brown University)===<br />
''Parabolic equations in time-varying domains''<br />
<br />
Abstract: I will present a recent result on the Dirichlet boundary value<br />
problem for parabolic equations in time-varying domains. The equations are<br />
in either divergence or non-divergence form with boundary blowup low-order<br />
coefficients. The domains satisfy an exterior measure condition.<br />
<br />
===Hao Jia (University of Chicago)===<br />
''Long time dynamics of energy critical defocusing wave equation with<br />
radial potential in 3+1 dimensions.''<br />
<br />
Abstract: We consider the long term dynamics of radial solution to the<br />
above mentioned equation. For general potential, the equation can have a<br />
unique positive ground state and a number of excited states. One can expect<br />
that some solutions might stay for very long time near excited states<br />
before settling down to an excited state of lower energy or the ground<br />
state. Thus the detailed dynamics can be extremely complicated. However<br />
using the ``channel of energy" inequality discovered by T.Duyckaerts,<br />
C.Kenig and F.Merle, we can show for generic potential, any radial solution<br />
is asymptotically the sum of a free radiation and a steady state as time<br />
goes to infinity. This provides another example of the power of ``channel<br />
of energy" inequality and the method of profile decompositions. I will<br />
explain the basic tools in some detail. Joint work with Baoping Liu and<br />
Guixiang Xu.<br />
<br />
===Alexander Pushnitski (King's College)===<br />
''An inverse spectral problem for Hankel operators''<br />
<br />
Abstract:<br />
I will discuss an inverse spectral problem for a certain class of Hankel<br />
operators. The problem appeared in the recent work by P.Gerard and S.Grellier as a<br />
step towards description of evolution in a model integrable non-dispersive<br />
equation. Several features of this inverse problem make it strikingly (and somewhat<br />
mysteriously) similar to an inverse problem for Sturm-Liouville operators. I will<br />
describe the available results for Hankel operators, emphasizing this similarity.<br />
This is joint work with Patrick Gerard (Orsay).<br />
<br />
===Ronghua Pan (Georgia Tech)===<br />
''Compressible Navier-Stokes-Fourier system with temperature dependent dissipation''<br />
<br />
Abstract: From its physical origin such as Chapman-Enskog or Sutherland, the viscosity and<br />
heat conductivity coefficients in compressible fluids depend on absolute temperature<br />
through power laws. The mathematical theory on the well-posedness and regularity on this<br />
setting is widely open. I will report some recent progress on this direction,<br />
with emphasis on the lower bound of temperature, and global existence of<br />
solutions in one or multiple dimensions. The relation between thermodynamics laws<br />
and Navier-Stokes-Fourier system will also be discussed. This talk is based on joint works<br />
with Junxiong Jia and Weizhe Zhang.<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Transonic shocks for Euler-Poisson system and related problems''<br />
<br />
Abstract: Euler-Poisson system models various physical phenomena<br />
including the propagation of electrons in submicron semiconductor<br />
devices and plasmas, and the biological transport of ions for channel<br />
proteins. I will explain difference between Euler system and<br />
Euler-Poisson system and mathematical difficulties arising due to this<br />
difference. And, recent results about subsonic flow and transonic flow<br />
for Euler-Poisson system will be presented. This talk is based on<br />
collaboration with Ben Duan and Chunjing Xie.<br />
<br />
<br />
===Mike Cullen (Met. Office, UK)===<br />
''Modelling the uncertainty in predicting large-scale atmospheric circulations''<br />
<br />
Abstract: This talk describes work on quantifying the uncertainty in climate<br />
models by using data assimilation techniques; and in estimating the properties<br />
of large-scale atmospheric flows by using asymptotic limit solutions.<br />
<br />
===Mohammad Ghomi(Math. Georgia Tech)===<br />
<br />
Abstract: We show that every smooth closed curve C immersed in Euclidean<br />
3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers<br />
P of pairs of parallel tangent lines, I of inflections (or points of<br />
vanishing curvature), and V of vertices (or points of vanishing torsion) of<br />
C. The proof, which employs curve shortening flow with surgery, is based on<br />
corresponding inequalities for the numbers of double points, singularities,<br />
and inflections of closed curves in the real projective plane and the sphere<br />
which intersect every closed geodesic. These findings extend some classical<br />
results in curve theory including works of Moebius, Fenchel, and Segre, which<br />
is also known as Arnold's ``tennis ball theorem".<br />
<br />
===Wei Xiang (Oxford)===<br />
<br />
Abstract: The vertical shock which initially separates two<br />
piecewise constant Riemann data, passes the wedge from left to right,<br />
then shock diffraction phenomena will occur and the incident shock<br />
becomes a transonic shock. Here we study this problem on nonlinear<br />
wave system as well as on potential flow equations. The existence and<br />
the optimal regularity across sonic circle of the solutions to this<br />
problem is established. The comparison of these two systems is<br />
discussed, and some related open problems are proposed.<br />
<br />
===Adrian Tudorascu (West Virginia University)===<br />
<br />
Abstract: <br />
This chalk & board presentation will start with an interesting motivation for<br />
these systems (the simplest of which was shown to model adhesion<br />
dynamics/ballistic aggregation by Zeldovich in 1970 in a paper published in<br />
"Astronomy & Astrophysics"). A general existence result for one-dimensional<br />
pressureless Euler/Euler-Poisson systems with/without viscosity will be<br />
discussed. This is achieved via entropy solutions for some appropriate scalar<br />
conservation laws; I will show that these solutions encode all the information<br />
necessary to obtain solutions for the pressureless systems. I will also discuss<br />
the stability and uniqueness of solutions, which is obtained via a contraction<br />
principle in the Wasserstein metric. This is based on joint work with T. Nguyen<br />
(U. Akron).<br />
<br />
===Diego Cordoba (Madrid)===<br />
<br />
Abstract: We consider the evolution of an interface generated between two immiscible, <br />
incompressible and irrotational fluids. Specifically we study the Muskat equation (the <br />
interface between oil and water in sand) and water wave equation (interface between water <br />
and vacuum). For both equations we will study well-posedness and the existence of smooth <br />
initial data for which the smoothness of the interface breaks down in finite time. We <br />
will also discuss some open problems.<br />
<br />
===Fausto Ferrari (Bologna)===<br />
''Semilinear PDEs and some symmetry properties of stable solutions''<br />
<br />
I will deal with stable solutions of semilinear elliptic PDE's and some of their<br />
symmetry's properties. Moreover, I will introduce some weighted Poincaré<br />
inequalities obtained by combining the notion of stable solution with the<br />
definition of weak solution.<br />
<br />
===Arshak Petrosyan (Purdue)===<br />
''Nonuniqueness in a free boundary problem from combustion''<br />
<br />
We consider a parabolic free boundary problem with a fixed gradient condition<br />
which serves as a simplified model for the propagation of premixed equidiffusional<br />
flames. We give a rigorous justification of an example due to J.L. V ́azquez that <br />
the initial data in the form of two circular humps leads to the nonuniqueness of limit <br />
solutions if the supports of the humps touch at the time of their maximal expansion.<br />
<br />
This is a joint work with Aaron Yip.<br />
<br />
<br />
===Changyou Wang (U. of Kentucky)===<br />
''Phase transition for higher dimensional wells''<br />
<br />
For a potential function <math>F</math> that has two global minimum sets consisting of two compact connected<br />
Riemannian submanifolds in <math style="vertical-align=100%" >\mathbb{R}^k</math>, we consider the singular perturbation problem:<br />
<br />
Minimizing <math>\int \left(|\nabla u|^2+\frac{1}{\epsilon^2} F(u)\right)</math> under given Dirichlet boundary data.<br />
<br />
I will discuss a recent joint work with F.H.Lin and X.B.Pan on the asymptotic, as the parameter <math>\epsilon</math><br />
tends to zero, in terms of the area of minimal hypersurface interfaces, the minimal connecting energy, and<br />
the energy of minimizing harmonic maps into the phase manifolds under both Dirichlet and partially free boundary<br />
data. Our results in particular addressed the static case of the so-called Keller-Rubinstein-Sternberg problem.<br />
<br />
===Philippe LeFloch (Paris VI)===<br />
''Kinetic relations for undercompressive shock waves and propagating phase boundaries''<br />
<br />
I will discuss the existence and properties of shock wave solutions to nonlinear hyperbolic systems that are small-scale dependent and, especially, contain undercompressive shock waves or propagating phase boundaries. Regularization-sensitive patterns often arise in continuum physics, especially in complex fluid flows. The so-called kinetic relation is introduced to characterize the correct dynamics of these nonclassical waves, and is tied to a higher-order regularization induced by a more complete model that takes into account additional small-scale physics. In the present lecture, I will especially explain the techniques of Riemann problems, Glimm-type scheme, and total variation functionals adapted to nonclassical shock waves. <br />
<br />
<br />
<br />
===Irina Mitrea===<br />
''Boundary Value Problems for Higher Order Differential Operators''<br />
<br />
As is well known, many phenomena in engineering and mathematical physics<br />
can be modeled by means of boundary value problems for a certain elliptic<br />
differential operator L in a domain D.<br />
<br />
When L is a differential operator of second order a variety of tools<br />
are available for dealing with such problems including boundary integral<br />
methods,<br />
variational methods, harmonic measure techniques, and methods based on<br />
classical<br />
harmonic analysis. The situation when the differential operator has higher order<br />
(as is the case for instance with anisotropic plate bending when one<br />
deals with<br />
fourth order) stands in sharp contrast with this as only fewer options<br />
could be<br />
successfully implemented. Alberto Calderon, one of the founders of the<br />
modern theory<br />
of Singular Integral Operators, has advocated in the seventies the use<br />
of layer potentials<br />
for the treatment of higher order elliptic boundary value problems.<br />
While the<br />
layer potential method has proved to be tremendously successful in the<br />
treatment<br />
of second order problems, this approach is insufficiently developed to deal<br />
with the intricacies of the theory of higher order operators. In fact,<br />
it is largely<br />
absent from the literature dealing with such problems.<br />
<br />
In this talk I will discuss recent progress in developing a multiple<br />
layer potential<br />
approach for the treatment of boundary value problems associated with<br />
higher order elliptic differential operators. This is done in a very<br />
general class<br />
of domains which is in the nature of best possible from the point of<br />
view of<br />
geometric measure theory. <br />
<br />
<br />
===Panagiota Daskalopoulos (Columbia U)===<br />
''Ancient solutions to geometric flows''<br />
<br />
We will discuss the clasification of ancient solutions to nonlinear geometric flows. <br />
It is well known that ancient solutions appear as blow up limits at a finite time <br />
singularity of the flow.<br />
Special emphasis will be given to the 2-dimensional Ricci flow.<br />
In this case we will show that ancient compact solution<br />
is either the Einstein (trivial) or one of the King-Rosenau solutions. <br />
<br />
===Maria Gualdani (UT Austin)===<br />
''A nonlinear diffusion model in mean-field games''<br />
<br />
We present an overview of mean-field games theory and show <br />
recent results on a free boundary value problem, which models <br />
price formation dynamics. <br />
In such model, the price is formed through a game among infinite number <br />
of agents. <br />
Existence and regularity results, as well as linear stability, will be shown.<br />
<br />
===Hiroshi Matano (Tokyo University)===<br />
''Traveling waves in a sawtoothed cylinder and their homogenization limit''<br />
<br />
My talk is concerned with a curvature-dependent motion of plane <br />
curves in a two-dimensional cylinder with spatially undulating<br />
boundary. In other words, the boundary has many bumps and we <br />
assume that the bumps are aligned in a spatially recurrent manner.<br />
<br />
The goal is to study how the average speed of the traveling wave <br />
depends on the geometry of the domain boundary. More specifically, <br />
we consider the homogenization problem as the boundary undulation <br />
becomes finer and finer, and determine the homogenization limit <br />
of the average speed and the limit profile of the traveling waves. <br />
Quite surprisingly, this homogenized speed depends only on the <br />
maximal opening angles of the domain boundary and no other <br />
geometrical features are relevant. <br />
<br />
Next we consider the special case where the boundary undulation <br />
is quasi-periodic with ''m'' independent frequencies. We show that <br />
the rate of convergence to the homogenization limit depends on<br />
this number ''m''.<br />
<br />
This is joint work with Bendong Lou and Ken-Ichi Nakamura.<br />
<br />
===Ian Tice (Brown University)===<br />
''Global well-posedness and decay for the viscous surface wave<br />
problem without surface tension''<br />
<br />
We study the incompressible, gravity-driven Navier-Stokes<br />
equations in three dimensional domains with free upper boundaries and<br />
fixed lower boundaries, in both the horizontally periodic and<br />
non-periodic settings. The effect of surface tension is not included.<br />
We employ a novel two-tier nonlinear energy method that couples the<br />
boundedness of certain high-regularity norms to the algebraic decay of<br />
lower-regularity norms. The algebraic decay allows us to balance the<br />
growth of the highest order derivatives of the free surface function,<br />
which then allows us to derive a priori estimates for solutions. We<br />
then prove local well-posedness in our energy space, which yields global<br />
well-posedness and decay. The novel LWP theory is established through<br />
the study of the linear Stokes problem in moving domains. This is joint<br />
work with Yan Guo.<br />
<br />
<br />
===Hoai Minh Nguyen (NYU-Courant Institute)===<br />
''Cloaking via change of variables for the Helmholtz equation''<br />
<br />
A region of space is cloaked for a class of measurements if observers<br />
are not only unaware of its contents, but also unaware of the presence<br />
of the cloak using such measurements. One approach to cloaking is the<br />
change of variables scheme introduced by Greenleaf, Lassas, and<br />
Uhlmann for electrical impedance tomography and by Pendry, Schurig,<br />
and Smith for the Maxwell equation. They used a singular change of<br />
variables which blows up a point into the cloaked region. To avoid<br />
this singularity, various regularized schemes have been proposed. In<br />
this talk I present results related to cloaking via change of<br />
variables for the Helmholtz equation using the natural regularized<br />
scheme introduced by Kohn, Shen, Vogelius, and Weintein, where the<br />
authors used a transformation which blows up a small ball instead of a<br />
point into the cloaked region. I will discuss the degree of<br />
invisibility for a finite range or the full range of frequencies, and<br />
the possibility of achieving perfect cloaking. If time permits, I will<br />
mention some results related to the wave equation.<br />
<br />
===Bing Wang (Princeton)===<br />
''The Kaehler Ricci flow on Fano manifold ''<br />
<br />
We show the convergence of the Kaehler Ricci flow on every 2-dimensional Fano manifold which admits big <math>\alpha_{\nu, 1}</math><br />
or <math>\alpha_{\nu, 2}</math> (Tian's invariants). Our method also works for 2-dimensional Fano orbifolds.<br />
Since Tian's invariants can be calculated by algebraic geometry method, our convergence theorem implies that one can find new Kaehler Einstein metrics<br />
on orbifolds by calculating Tian's invariants.<br />
An essential part of the proof is to confirm the Hamilton-Tian conjecture in complex dimension 2.<br />
<br />
===Francois Hamel (Marseille)===<br />
''Optimization of eigenvalues of non-symmetric elliptic operators''<br />
<br />
The talk is concerned with various optimization results for the principal eigenvalues of general second-order elliptic operators in divergence form with Dirichlet boundary condition in bounded domains of <math>R^n</math>. To each operator in a given domain, we can associate a radially symmetric operator in the ball with the same Lebesgue measure, with a smaller principal eigenvalue. The constraints on the coefficients are of integral, pointwise or geometric types. In particular, we generalize the Rayleigh-Faber-Krahn inequality for the principal eigenvalue of the Dirichlet Laplacian. The proofs use a new symmetrization technique, different from the Schwarz symmetrization. This talk is based on a joint work with N. Nadirashvili and E. Russ.<br />
<br />
===Juraj Foldes (Vanderbilt)===<br />
''Symmetry properties of parabolic problems and their applications''<br />
<br />
Positive solutions of nonlinear parabolic problems can have a very complex behavior. However, assuming certain symmetry conditions, it is possible to prove that the solutions converge to the space of symmetric functions. We show that this property is 'stable'; more specifically if the symmetry conditions are replaced by asymptotically symmetric ones, the solutions still approach the space of symmetric functions. As an application, we show new results on convergence of solutions to a single equilibrium.<br />
<br />
===Alexey Cheskidov (UIC)===<br />
''Navier-Stokes and Euler equations: a unified approach to the problem of blow-up''<br />
<br />
The problems of blow-up for Navier-Stokes and Euler equations <br />
have been extensively studied for decades using different techniques. <br />
Motivated by Kolmogorov's theory of turbulence, we present a new unified <br />
approach to the blow-up problem for the equations of incompressible <br />
fluid motion. In particular, we present a new regularity criterion which <br />
is weaker than the Beale-Kato-Majda condition in the inviscid case, and <br />
weaker than every Ladyzhenskaya-Prodi-Serrin condition in the viscous case.<br />
<br />
===Takis Souganidis (Chicago)===<br />
''Stochastic homogenization of the G-equation''<br />
<br />
The G-equation is a Hamilton-Jacobi equation, of level-set-type, which is used as a model in turbulent combustion. In the lecture I will present recent joint work with Pierre Cardaliaguet about the homogenization of the G-equation set in random media, when the problem is not coercive and, hence, falls outside the scope of the theory of stochastic homogenization.<br />
<br />
===Scott Armstrong (UW-Madison)===<br />
''Partial regularity for fully nonlinear elliptic equations''<br />
<br />
I will present some regularity results for (nonconvex) fully nonlinear equations. Such equations do not possess smooth solutions, but in joint work with Silvestre and Smart we show that the Hausdorff dimension of the singular set is less than the ambient dimension. Using an argument with a similar flavor, we prove (jointly with Silvestre) a unique continuation result for such equations.<br />
<br />
===Russell Schwab (Carnegie Mellon)===<br />
''On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)''<br />
<br />
Despite much recent (and not so recent) attention to solutions of integro-differential equations of elliptic type, it is surprising that a fundamental result such as a comparison theorem which can deal with only measure theoretic norms of the right hand side of the equation (L-n and L-infinity) has gone unexplored. For the case of second order equations this result is known as the Aleksandrov-Bakelman-Pucci estimate (and dates back to circa 1960s), which says that for supersolutions of uniformly elliptic equation Lu=f, the supremum of u is controlled by the L-n norm of f (n being the underlying dimension of the domain). We discuss extensions of this estimate to fully nonlinear integro-differential equations and present a recent result in this direction. (Joint with Nestor Guillen, available at arXiv:1101.0279v3 [math.AP])<br />
<br />
===Valentin Ovsienko (University of Lyon)===<br />
''The pentagram map and generalized friezes of Coxeter''<br />
<br />
The pentagram map is a discrete integrable system on the moduli space of n-gons in the projective plane (which is a close relative of the moduli space of genus 0 curves with n marked points). The most interesting properties of the pentagram map is its relations to the theory of cluster algebras and to the classical integrable systems (such as the Boussinesq equation). I will talk of the recent results proving the integrability as well as of the algebraic and arithmetic properties of the pentagram map.<br />
In particular, I will introduce the space of 2-frieze patterns generalizing that of the classical Coxeter friezes and define the structure of cluster manifold on this space. The talk is based on joint works with Sophie Morier-Genoud, Richard Schwartz and Serge Tabachnikov.<br />
<br />
<br />
===Adrian Tudorascu (West Virginia University)===<br />
''Weak Lagrangian solutions for the Semi-Geostrophic system in physical space''<br />
<br />
Proposed as a simplification for the Boussinesq system in a special regime, the Semi-Geostrophic (SG) system is used by metereologists to model how fronts arise in large scale weather patterns. In spite of significant progress achieved in the analysis of the SG in dual space (i.e. the system obtained from the SG by a special change of variables), there are no existence results on the SG in physical space. We shall argue that weak (Eulerian) solutions for the Semi-Geostrophic system in physical space exhibiting some mild regularity in time cannot yield point masses in the dual space. However, such solutions are physically relevant to the model. Thus, we shall discuss a natural generalization of Cullen & Feldman's weak Lagrangian solutions in the physical space to include the possibility of singular measures in dual space. We have proved existence of such solutions in the case of discrete measures in dual space. The talk is based on joint work with M. Feldman.<br />
<br />
===James Nolen (Duke)===<br />
''Normal approximation for a random elliptic PDE''<br />
<br />
I will talk about solutions to an elliptic PDE with conductivity coefficient that varies randomly with respect to the spatial variable. It has been known for some time that homogenization may occur when the coefficients are scaled suitably. Less is known about fluctuations of the solution around its mean behavior. This talk is about the energy dissipation rate, which is a quadratic functional of the solution and a bulk property of the random material. For a finite random sample of the material, this quantity is random. In the limit of large sample size it converges to a deterministic constant. I will describe a central limit theorem: the probability law of the energy dissipation rate is very close to that of a normal random variable having the same mean and variance. I'll give an error estimate for this approximation in total variation.<br />
<br />
===Betsy Stovall (UCLA)===<br />
<br />
We will discuss recent work concerning the cubic Klein--Gordon equation<br />
u_{tt} - \Delta u + u \pm u^3 = 0<br />
in two space dimensions with real valued initial data in the energy space, u(0) in H^1, u_t(0) in L^2. We show that in the defocusing case, solutions<br />
are global and have finite L^4 norm (in space and time). In the focusing case, we characterize the dichotomy between such behavior and finite time<br />
blowup for initial data having energy less than that of the ground state. In this talk, we will pay particular attention to connections with certain<br />
questions arising in harmonic analysis. <br />
<br />
This is joint work with Rowan Killip and Monica Visan.<br />
<br />
===Charles Smart (MIT)===<br />
''Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian''<br />
<br />
A classical theorem of Kirszbraun states that any Lipschitz function $f : A \to \R^m$ defined on a set $A \subseteq \R^n$ can be extended to all of $\R^n$ without increasing the Lipschitz constant. The search for a canonical such extension leads one to the notion of optimal Lipschitz extension. I will discuss joint work with Evans on the regularity of optimal Lipschitz extensions in the scalar $m = 1$ case and joint work with Sheffield on the vector valued $m > 1$ case.<br />
<br />
<br />
===Yao Yao (UCLA)===<br />
''Degenerate diffusion with nonlocal aggregation: behavior of solutions''<br />
<br />
The Patlak-Keller-Segel (PKS) equation models the collective motion of<br />
cells which are attracted by a self-emitted chemical substance. While the<br />
global well-posedness and finite-time blow up criteria are well known, the<br />
asymptotic behaviors of solutions are not completely clear. In this talk I<br />
will present some results on the asymptotic behavior of solutions when<br />
there is global existence. The key tools used in the paper are<br />
maximum-principle type arguments as well as estimates on mass concentration<br />
of solutions. This is a joint work with Inwon Kim.<br />
<br />
===Xuan Hien Nguyen (Iowa State)===<br />
<br />
''Gluing constructions for solitons and self-shrinkers under mean curvature flow''<br />
<br />
In the 1990s, Kapouleas and Traizet constructed new examples of minimal surfaces by desingularizing the intersection of existing ones with Scherk surfaces. Using this idea, one can find new examples of self-translating solutions for the mean curvature flow asymptotic at infinity to a finite family of grim reaper cylinders in general position. Recently, it has been shown that it is possible to desingularize the intersection of a sphere and a plane to obtain a family of self-shrinkers under mean curvature flow. I will discuss the main steps and difficulties for these gluing constructions, as well as open problems.<br />
<br />
===Nestor Guillen (UCLA)===<br />
<br />
We consider the Monge-Kantorovich problem, which consists in<br />
transporting a given measure into another "target" measure in a way<br />
that minimizes the total cost of moving each unit of mass to its new<br />
location. When the transport cost is given by the square of the<br />
distance between two points, the optimal map is given by a convex<br />
potential which solves the Monge-Ampère equation, in general, the<br />
solution is given by what is called a c-convex potential. In recent<br />
work with Jun Kitagawa, we prove local Holder estimates of optimal<br />
transport maps for more general cost functions satisfying a<br />
"synthetic" MTW condition, in particular, the proof does not really<br />
use the C^4 assumption made in all previous works. A similar result<br />
was recently obtained by Figalli, Kim and McCann using different<br />
methods and assuming strict convexity of the target.<br />
<br />
===Charles Smart (MIT)===<br />
<br />
''PDE methods for the Abelian sandpile''<br />
<br />
Abstract: The Abelian sandpile growth model is a deterministic<br />
diffusion process for chips placed on the $d$-dimensional integer<br />
lattice. One of the most striking features of the sandpile is that it<br />
appears to produce terminal configurations converging to a peculiar<br />
lattice. One of the most striking features of the sandpile is that it<br />
appears to produce terminal configurations converging to a peculiar<br />
fractal limit when begun from increasingly large stacks of chips at<br />
the origin. This behavior defied explanation for many years until<br />
viscosity solution theory offered a new perspective. This is joint<br />
work with Lionel Levine and Wesley Pegden.<br />
<br />
===Vlad Vicol (University of Chicago)===<br />
<br />
Title: Shape dependent maximum principles and applications<br />
<br />
Abstract: We present a non-linear lower bound for the fractional Laplacian, when<br />
evaluated at extrema of a function. Applications to the global well-posedness of active<br />
scalar equations arising in fluid dynamics are discussed. This is joint work with P.<br />
Constantin.<br />
<br />
<br />
===Jiahong Wu (Oklahoma State)===<br />
<br />
"The 2D Boussinesq equations with partial dissipation"<br />
<br />
The Boussinesq equations concerned here model geophysical flows such<br />
as atmospheric fronts and ocean circulations. Mathematically the 2D Boussinesq<br />
equations serve as a lower-dimensional model of the 3D hydrodynamics<br />
equations. In fact, the 2D Boussinesq equations retain some key features<br />
of the 3D Euler and the Navier-Stokes equations such as the vortex stretching<br />
mechanism. The global regularity problem on the 2D Boussinesq equations<br />
with partial dissipation has attracted considerable attention in the last few years.<br />
In this talk we will summarize recent results on various cases of partial dissipation,<br />
present the work of Cao and Wu on the 2D Boussinesq equations with vertical<br />
dissipation and vertical thermal diffusion, and explain the work of Chae and Wu on<br />
the critical Boussinesq equations with a logarithmically singular velocity.<br />
<br />
<br />
===Joana Oliveira dos Santos Amorim (Universit\'e Paris Dauphine)===<br />
<br />
"A geometric look on Aubry-Mather theory and a theorem of Birkhoff"<br />
<br />
Given a Tonelli Hamiltonian $H:T^*M \lto \Rm$ in the cotangent bundle of a compact manifold $M$, <br />
we can study its dynamic using the Aubry and Ma\~n\'e sets defined by Mather. <br />
In this talk we will explain their importance and give a new geometric definition <br />
which allows us to understand their property of symplectic invariance. <br />
Moreover, using this geometric definition, we will show that an exact <br />
Lipchitz Lagrangian manifold isotopic to a graph which is invariant <br />
by the flow of a Tonelli Hamiltonian is itself a graph. <br />
This result, in its smooth form, was a conjecture of Birkhoff.<br />
<br />
<br />
===Gui-Qiang Chen (Oxford) ===<br />
<br />
"Nonlinear Partial Differential Equations of Mixed Type"<br />
<br />
Many nonlinear partial differential equations arising in mechanics and geometry naturally are of mixed hyperbolic-elliptic type. The solution of some longstanding fundamental problems in these areas greatly requires a deep understanding of such nonlinear partial differential equations of mixed type. Important examples include<br />
shock reflection-diffraction problems in fluid mechanics (the Euler equations), isometric embedding problems in in differential geometry (the Gauss-Codazzi-Ricci equations),<br />
among many others. In this talk we will present natural connections of nonlinear partial differential equations with these longstanding problems and will discuss some recent developments in the analysis of these nonlinear equations through the examples with emphasis on identifying/developing mathematical approaches, ideas, and techniques to deal with the mixed-type problems. Further trends, perspectives, and open problems in this direction will also be addressed.<br />
This talk will be based mainly on the joint work correspondingly with Mikhail Feldman, Marshall Slemrod, as well as Myoungjean Bae and Dehua Wang. <br />
<br />
<br />
===Jacob Glenn-Levin (UT Austin)===<br />
<br />
We consider the Boussinesq equations, which may be thought of as inhomogeneous, <br />
incompressible Euler equations, where the inhomogeneous term is a scalar quantity, <br />
typically density or temperature, governed by a convection-diffusion equation. I will <br />
discuss local- and global-in-time well-posedness results for the incompressible 2D <br />
Boussinesq equations, assuming the density equation has nonzero diffusion and that the <br />
initial data belongs in a Besov-type space.<br />
<br />
===Bing Wang (UW Madison)===<br />
''On the regularity of limit space''<br />
<br />
This is a joint work with Gang Tian. <br />
In this talk, we will discuss how to improve regularity of the limit space by Ricci flow.<br />
We study the structure of the limit space of a sequence of almost Einstein<br />
manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such<br />
manifolds are the initial manifolds of some normalized Ricci flows whose scalar<br />
curvatures are almost constants over space-time in the L1-sense, Ricci curvatures<br />
are bounded from below at the initial time. Under the non-collapsed condition,<br />
we show that the limit space of a sequence of almost Einstein manifolds has most<br />
properties which is known for the limit space of Einstein manifolds. As applications,<br />
we can apply our structure results to study the properties of K¨ahler manifolds.<br />
<br />
<br />
===Peter Polacik (University of Minnesota)===<br />
'' Exponential separation between positive and sign-changing solutions and its applications''<br />
<br />
In linear nonautonomous second-order parabolic equations, the exponential separation refers to the exponential decay of any sign-changing solution relative to any positive solution. In this lecture, after summarizing key results on exponential separation, we show how it can be effectively used in studies of some nonlinear parabolic problems.<br />
<br />
<br />
<br />
===Kyudong Choi (UW Madison)===<br />
''Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations''<br />
<br />
We study weak solutions of the 3D Navier-Stokes equations with L^2 initial data. We <br />
prove that k-th derivative of weak solutions is locally integrable in space-time for any <br />
real k such that 1 < k < 3. Up to now, only the second derivative was known to be locally <br />
integrable by standard parabolic regularization. We also present sharp estimates of those <br />
quantities in weak-L^{4/(k+1)} locally. These estimates depend only on the L^2 norm of <br />
the initial data and on the domain of integration. Moreover, they are valid even for k >= <br />
3 as long as we have a smooth solution. The proof uses a standard approximation of <br />
Navier-Stokes from Leray and a blow-up techniques. The local study is based on De Giorgi <br />
techniques with a new pressure decomposition. To handle the non-locality of fractional <br />
Laplacians, Hardy space and Maximal functions are introduced. This is joint work with A. <br />
Vasseur.<br />
<br />
<br />
===Yao Yao (UW Madison)===<br />
''Confinement for nonlocal interaction equation with repulsive-attractive kernels''<br />
<br />
In this talk, we consider sets of particles interacting pairwise via a potential W, which is repulsive at short distances, but attractive at long distances. The main question we consider is whether an initially compactly supported configuration remains compactly supported for all times, regardless of the number of particles. We improved the sufficient conditions on the potential W for the above confinement property to hold. This is a joint work with Jose Carrillo and Daniel Balague.<br />
<br />
<br />
<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Transonic shocks for Euler-Poisson system and related problems''<br />
<br />
Abstract: Euler-Poisson system models various physical phenomena<br />
including the propagation of electrons in submicron semiconductor<br />
devices and plasmas, and the biological transport of ions for channel<br />
proteins. I will explain difference between Euler system and<br />
Euler-Poisson system and mathematical difficulties arising due to this<br />
difference. And, recent results about subsonic flow and transonic flow<br />
for Euler-Poisson system will be presented. This talk is based on<br />
collaboration with Ben Duan and Chunjing Xie.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17615Fall 2019-Spring 20202019-08-06T09:40:10Z<p>Ckim: </p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | TBA ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17609Fall 2019-Spring 20202019-08-04T21:01:57Z<p>Ckim: </p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | TBA ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17350Fall 2019-Spring 20202019-04-19T21:44:00Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17349Fall 2019-Spring 20202019-04-19T21:43:30Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| J Jang (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17171PDE Geometric Analysis seminar2019-03-15T15:36:55Z<p>Ckim: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | Stability of vacuum for the Landau equation with moderately soft potentials ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria. ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
===Jonathan Luk===<br />
<br />
Title: Stability of vacuum for the Landau equation with moderately soft potentials<br />
<br />
Abstract: Consider the Landau equation with moderately soft potentials in the whole space. We prove that sufficiently small and localized regular initial data give rise to unique global-in-time smooth solutions. Moreover, the solutions approach that of the free transport equation as $t\to +\infty$. This is the first stability of vacuum result for a binary collisional kinetic model featuring a long-range interaction.<br />
<br />
<br />
===Jiaxin Jin===<br />
<br />
Title: Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria.<br />
<br />
Abstract: We first analyze a three-species system with boundary equilibria in some stoichiometric classes and study the rate of convergence to the complex balanced equilibrium. Then we prove similar results on the convergence to the positive equilibrium for a fairly general two-species reversible reaction-diffusion network with boundary equilibria.<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17170PDE Geometric Analysis seminar2019-03-15T15:36:28Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | Stability of vacuum for the Landau equation with moderately soft potentials ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria. ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
===Jonathan Luk===<br />
<br />
Title: Stability of vacuum for the Landau equation with moderately soft potentials<br />
<br />
Abstract: Consider the Landau equation with moderately soft potentials in the whole space. We prove that sufficiently small and localized regular initial data give rise to unique global-in-time smooth solutions. Moreover, the solutions approach that of the free transport equation as $t\to +\infty$. This is the first stability of vacuum result for a binary collisional kinetic model featuring a long-range interaction.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17169Fall 2019-Spring 20202019-03-15T14:46:10Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17168Fall 2019-Spring 20202019-03-15T14:42:47Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ | ]]<br />
| <br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17167Fall 2019-Spring 20202019-03-15T14:41:40Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17166Fall 2019-Spring 20202019-03-15T13:51:19Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17165Fall 2019-Spring 20202019-03-14T23:53:11Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 10<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 17<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17164Fall 2019-Spring 20202019-03-14T23:52:37Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
| <br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
| <br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
| <br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
| <br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 10<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 17<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17163Fall 2019-Spring 20202019-03-14T23:51:37Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
| <br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
| <br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Jan 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 3<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 10<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 17<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17162Fall 2019-Spring 20202019-03-14T23:48:07Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
| <br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
| <br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Jan 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17114PDE Geometric Analysis seminar2019-03-05T23:26:56Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | Stability of vacuum for the Landau equation with moderately soft potentials ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |TBA ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
===Jonathan Luk===<br />
<br />
Title: Stability of vacuum for the Landau equation with moderately soft potentials<br />
<br />
Abstract: Consider the Landau equation with moderately soft potentials in the whole space. We prove that sufficiently small and localized regular initial data give rise to unique global-in-time smooth solutions. Moreover, the solutions approach that of the free transport equation as $t\to +\infty$. This is the first stability of vacuum result for a binary collisional kinetic model featuring a long-range interaction.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17113PDE Geometric Analysis seminar2019-03-05T23:26:35Z<p>Ckim: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |TBA ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
===Jonathan Luk===<br />
<br />
Title: Stability of vacuum for the Landau equation with moderately soft potentials<br />
<br />
Abstract: Consider the Landau equation with moderately soft potentials in the whole space. We prove that sufficiently small and localized regular initial data give rise to unique global-in-time smooth solutions. Moreover, the solutions approach that of the free transport equation as $t\to +\infty$. This is the first stability of vacuum result for a binary collisional kinetic model featuring a long-range interaction.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=17029Colloquia2019-02-22T14:59:08Z<p>Ckim: /* Abstracts */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
==Spring 2019==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Jan 25 '''Room 911'''<br />
| [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW<br />
|[[#Beata Randrianantoanina (Miami University Ohio) | Some nonlinear problems in the geometry of Banach spaces and their applications ]]<br />
| Tullia Dymarz<br />
|<br />
|-<br />
|Jan 30 '''Wednesday'''<br />
| Talk rescheduled to Feb 15<br />
|<br />
|-<br />
|Jan 31 '''Thursday'''<br />
| Talk rescheduled to Feb 13<br />
|<br />
|-<br />
|Feb 1<br />
| Talk cancelled due to weather<br />
|<br />
| <br />
|<br />
|-<br />
|Feb 5 '''Tuesday, VV 911'''<br />
| [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University)<br />
|[[#Alexei Poltoratski (Texas A&M)| Completeness of exponentials: Beurling-Malliavin and type problems ]]<br />
| Denisov<br />
|<br />
|-<br />
|Feb 6 '''Wednesday, room 911'''<br />
| [https://lc-tsai.github.io/ Li-Cheng Tsai] (Columbia University)<br />
|[[#Li-Cheng Tsai (Columbia University)| When particle systems meet PDEs ]]<br />
| Anderson<br />
|<br />
|-<br />
|Feb 8<br />
| [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)<br />
|[[#Aaron Naber (Northwestern) | A structure theory for spaces with lower Ricci curvature bounds ]]<br />
| Street<br />
|<br />
|-<br />
|Feb 11 '''Monday'''<br />
| [https://www2.bc.edu/david-treumann/materials.html David Treumann] (Boston College)<br />
|[[#David Treumann (Boston College) | Twisting things in topology and symplectic topology by pth powers ]]<br />
| Caldararu<br />
|<br />
|-<br />
| Feb 13 '''Wednesday'''<br />
| [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M)<br />
|[[#Dean Baskin (Texas A&M) | Radiation fields for wave equations ]]<br />
| Street<br />
<br />
|-<br />
| Feb 15 <br />
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)<br />
| [[#Lillian Pierce (Duke University) | Short character sums ]]<br />
| Boston and Street<br />
|<br />
|-<br />
|Feb 22<br />
| [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State)<br />
|[[#Angelica Cueto (The Ohio State University)| Lines on cubic surfaces in the tropics ]]<br />
| Erman and Corey<br />
|<br />
|-<br />
|March 4<br />
| [http://www-users.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) <br />
|[[# Vladimir Sverak (Minnesota) | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|<br />
|-<br />
|March 8<br />
| [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State)<br />
|[[# TBA| TBA ]]<br />
| Erman<br />
|<br />
|-<br />
|March 15<br />
| Maksym Radziwill (Caltech)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|March 29<br />
| Jennifer Park (OSU)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|April 5<br />
| Ju-Lee Kim (MIT)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 12<br />
| Eviatar Procaccia (TAMU)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 19<br />
| [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)<br />
|[[# TBA| TBA ]]<br />
| Jean-Luc<br />
|<br />
|-<br />
|April 22 '''Monday'''<br />
| [https://justinh.su Justin Hsu] (Madison)<br />
|[[# TBA| TBA ]]<br />
| Lempp<br />
|<br />
|-<br />
|April 26<br />
| [https://www.brown.edu/academics/applied-mathematics/faculty/kavita-ramanan/home Kavita Ramanan] (Brown University)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
|May 3<br />
| Tomasz Przebinda (Oklahoma)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Beata Randrianantoanina (Miami University Ohio)===<br />
<br />
Title: Some nonlinear problems in the geometry of Banach spaces and their applications.<br />
<br />
Abstract: Nonlinear problems in the geometry of Banach spaces have been studied since the inception of the field. In this talk I will outline some of the history, some of modern applications, and some open directions of research. The talk will be accessible to graduate students of any field of mathematics.<br />
<br />
===Lillian Pierce (Duke University)===<br />
<br />
Title: Short character sums <br />
<br />
Abstract: A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.<br />
<br />
===Angelica Cueto (The Ohio State University)===<br />
Title: Lines on cubic surfaces in the tropics<br />
<br />
Abstract: Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-know statement <i>any smooth surface of degree three in P^3 contains exactly 27 lines</i> is known to be false tropically. Work of Vigeland from 2007 provides examples of tropical cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in TP^3.<br />
<br />
In this talk I will explain how to correct this pathology by viewing the surface as a del Pezzo cubic and considering its embedding in P^44 via its anticanonical bundle. The combinatorics of the root system of type E_6 and a tropical notion of convexity will play a central role in the construction. This is joint work in progress with Anand Deopurkar.<br />
<br />
===David Treumann (Boston College)===<br />
<br />
Title: Twisting things in topology and symplectic topology by pth powers<br />
<br />
Abstract: There's an old and popular analogy between circles and finite fields. I'll describe some constructions you can make in Lagrangian Floer theory and in microlocal sheaf theory by taking this analogy extremely literally, the main ingredient is an "F-field." An F-field on a manifold M is a local system of algebraically closed fields of characteristic p. When M is symplectic, maybe an F-field should remind you of a B-field, it can be used to change the Fukaya category in about the same way. On M = S^1 times R^3, this version of the Fukaya category is related to Deligne-Lusztig theory, and I found something like a cluster structure on the Deligne-Lusztig pairing varieties by studying it. On M = S^1 times S^1, Yanki Lekili and I have found that this version of the Fukaya category is related to the equal-characteristic version of the Fargues-Fontaine curve; the relationship is homological mirror symmetry.<br />
<br />
===Dean Baskin (Texas A&M)===<br />
<br />
Title: Radiation fields for wave equations<br />
<br />
Abstract: Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.<br />
<br />
===Jianfeng Lu (Duke University)===<br />
<br />
Title: Density fitting: Analysis, algorithm and applications<br />
<br />
Abstract: Density fitting considers the low-rank approximation of pair products of eigenfunctions of Hamiltonian operators. It is a very useful tool with many applications in electronic structure theory. In this talk, we will discuss estimates of upper bound of the numerical rank of the pair products of eigenfunctions. We will also introduce the interpolative separable density fitting (ISDF) algorithm, which reduces the computational scaling of the low-rank approximation and can be used for efficient algorithms for electronic structure calculations. Based on joint works with Chris Sogge, Stefan Steinerberger, Kyle Thicke, and Lexing Ying.<br />
<br />
===Alexei Poltoratski (Texas A&M)===<br />
<br />
Title: Completeness of exponentials: Beurling-Malliavin and type problems<br />
<br />
Abstract: This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both<br />
problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin<br />
problem was solved in the early 1960s and I will present its classical solution along with modern generalizations<br />
and applications. I will then discuss history and recent progress in the type problem, which stood open for<br />
more than 70 years.<br />
<br />
===Li-Cheng Tsai (Columbia University)===<br />
<br />
Title: When particle systems meet PDEs<br />
<br />
Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems.<br />
<br />
===Aaron Naber (Northwestern)===<br />
<br />
Title: A structure theory for spaces with lower Ricci curvature bounds.<br />
<br />
Abstract: One should view manifolds (M^n,g) with lower Ricci curvature bounds as being those manifolds with a well behaved analysis, a point which can be rigorously stated. It thus becomes a natural question, how well behaved or badly behaved can such spaces be? This is a nonlinear analogue to asking how degenerate can a subharmonic or plurisubharmonic function look like. In this talk we give an essentially sharp answer to this question. The talk will require little background, and our time will be spent on understanding the basic statements and examples. The work discussed is joint with Cheeger, Jiang and with Li.<br />
<br />
<br />
===Vladimir Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=17028Colloquia2019-02-22T14:58:36Z<p>Ckim: /* Spring 2019 */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
==Spring 2019==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Jan 25 '''Room 911'''<br />
| [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW<br />
|[[#Beata Randrianantoanina (Miami University Ohio) | Some nonlinear problems in the geometry of Banach spaces and their applications ]]<br />
| Tullia Dymarz<br />
|<br />
|-<br />
|Jan 30 '''Wednesday'''<br />
| Talk rescheduled to Feb 15<br />
|<br />
|-<br />
|Jan 31 '''Thursday'''<br />
| Talk rescheduled to Feb 13<br />
|<br />
|-<br />
|Feb 1<br />
| Talk cancelled due to weather<br />
|<br />
| <br />
|<br />
|-<br />
|Feb 5 '''Tuesday, VV 911'''<br />
| [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University)<br />
|[[#Alexei Poltoratski (Texas A&M)| Completeness of exponentials: Beurling-Malliavin and type problems ]]<br />
| Denisov<br />
|<br />
|-<br />
|Feb 6 '''Wednesday, room 911'''<br />
| [https://lc-tsai.github.io/ Li-Cheng Tsai] (Columbia University)<br />
|[[#Li-Cheng Tsai (Columbia University)| When particle systems meet PDEs ]]<br />
| Anderson<br />
|<br />
|-<br />
|Feb 8<br />
| [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)<br />
|[[#Aaron Naber (Northwestern) | A structure theory for spaces with lower Ricci curvature bounds ]]<br />
| Street<br />
|<br />
|-<br />
|Feb 11 '''Monday'''<br />
| [https://www2.bc.edu/david-treumann/materials.html David Treumann] (Boston College)<br />
|[[#David Treumann (Boston College) | Twisting things in topology and symplectic topology by pth powers ]]<br />
| Caldararu<br />
|<br />
|-<br />
| Feb 13 '''Wednesday'''<br />
| [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M)<br />
|[[#Dean Baskin (Texas A&M) | Radiation fields for wave equations ]]<br />
| Street<br />
<br />
|-<br />
| Feb 15 <br />
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)<br />
| [[#Lillian Pierce (Duke University) | Short character sums ]]<br />
| Boston and Street<br />
|<br />
|-<br />
|Feb 22<br />
| [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State)<br />
|[[#Angelica Cueto (The Ohio State University)| Lines on cubic surfaces in the tropics ]]<br />
| Erman and Corey<br />
|<br />
|-<br />
|March 4<br />
| [http://www-users.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) <br />
|[[# Vladimir Sverak (Minnesota) | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|<br />
|-<br />
|March 8<br />
| [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State)<br />
|[[# TBA| TBA ]]<br />
| Erman<br />
|<br />
|-<br />
|March 15<br />
| Maksym Radziwill (Caltech)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|March 29<br />
| Jennifer Park (OSU)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|April 5<br />
| Ju-Lee Kim (MIT)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 12<br />
| Eviatar Procaccia (TAMU)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 19<br />
| [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)<br />
|[[# TBA| TBA ]]<br />
| Jean-Luc<br />
|<br />
|-<br />
|April 22 '''Monday'''<br />
| [https://justinh.su Justin Hsu] (Madison)<br />
|[[# TBA| TBA ]]<br />
| Lempp<br />
|<br />
|-<br />
|April 26<br />
| [https://www.brown.edu/academics/applied-mathematics/faculty/kavita-ramanan/home Kavita Ramanan] (Brown University)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
|May 3<br />
| Tomasz Przebinda (Oklahoma)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Beata Randrianantoanina (Miami University Ohio)===<br />
<br />
Title: Some nonlinear problems in the geometry of Banach spaces and their applications.<br />
<br />
Abstract: Nonlinear problems in the geometry of Banach spaces have been studied since the inception of the field. In this talk I will outline some of the history, some of modern applications, and some open directions of research. The talk will be accessible to graduate students of any field of mathematics.<br />
<br />
===Lillian Pierce (Duke University)===<br />
<br />
Title: Short character sums <br />
<br />
Abstract: A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.<br />
<br />
===Angelica Cueto (The Ohio State University)===<br />
Title: Lines on cubic surfaces in the tropics<br />
<br />
Abstract: Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-know statement <i>any smooth surface of degree three in P^3 contains exactly 27 lines</i> is known to be false tropically. Work of Vigeland from 2007 provides examples of tropical cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in TP^3.<br />
<br />
In this talk I will explain how to correct this pathology by viewing the surface as a del Pezzo cubic and considering its embedding in P^44 via its anticanonical bundle. The combinatorics of the root system of type E_6 and a tropical notion of convexity will play a central role in the construction. This is joint work in progress with Anand Deopurkar.<br />
<br />
===David Treumann (Boston College)===<br />
<br />
Title: Twisting things in topology and symplectic topology by pth powers<br />
<br />
Abstract: There's an old and popular analogy between circles and finite fields. I'll describe some constructions you can make in Lagrangian Floer theory and in microlocal sheaf theory by taking this analogy extremely literally, the main ingredient is an "F-field." An F-field on a manifold M is a local system of algebraically closed fields of characteristic p. When M is symplectic, maybe an F-field should remind you of a B-field, it can be used to change the Fukaya category in about the same way. On M = S^1 times R^3, this version of the Fukaya category is related to Deligne-Lusztig theory, and I found something like a cluster structure on the Deligne-Lusztig pairing varieties by studying it. On M = S^1 times S^1, Yanki Lekili and I have found that this version of the Fukaya category is related to the equal-characteristic version of the Fargues-Fontaine curve; the relationship is homological mirror symmetry.<br />
<br />
===Dean Baskin (Texas A&M)===<br />
<br />
Title: Radiation fields for wave equations<br />
<br />
Abstract: Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.<br />
<br />
===Jianfeng Lu (Duke University)===<br />
<br />
Title: Density fitting: Analysis, algorithm and applications<br />
<br />
Abstract: Density fitting considers the low-rank approximation of pair products of eigenfunctions of Hamiltonian operators. It is a very useful tool with many applications in electronic structure theory. In this talk, we will discuss estimates of upper bound of the numerical rank of the pair products of eigenfunctions. We will also introduce the interpolative separable density fitting (ISDF) algorithm, which reduces the computational scaling of the low-rank approximation and can be used for efficient algorithms for electronic structure calculations. Based on joint works with Chris Sogge, Stefan Steinerberger, Kyle Thicke, and Lexing Ying.<br />
<br />
===Alexei Poltoratski (Texas A&M)===<br />
<br />
Title: Completeness of exponentials: Beurling-Malliavin and type problems<br />
<br />
Abstract: This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both<br />
problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin<br />
problem was solved in the early 1960s and I will present its classical solution along with modern generalizations<br />
and applications. I will then discuss history and recent progress in the type problem, which stood open for<br />
more than 70 years.<br />
<br />
===Li-Cheng Tsai (Columbia University)===<br />
<br />
Title: When particle systems meet PDEs<br />
<br />
Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems.<br />
<br />
===Aaron Naber (Northwestern)===<br />
<br />
Title: A structure theory for spaces with lower Ricci curvature bounds.<br />
<br />
Abstract: One should view manifolds (M^n,g) with lower Ricci curvature bounds as being those manifolds with a well behaved analysis, a point which can be rigorously stated. It thus becomes a natural question, how well behaved or badly behaved can such spaces be? This is a nonlinear analogue to asking how degenerate can a subharmonic or plurisubharmonic function look like. In this talk we give an essentially sharp answer to this question. The talk will require little background, and our time will be spent on understanding the basic statements and examples. The work discussed is joint with Cheeger, Jiang and with Li.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=17027Colloquia2019-02-22T14:58:19Z<p>Ckim: /* Spring 2019 */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
==Spring 2019==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Jan 25 '''Room 911'''<br />
| [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW<br />
|[[#Beata Randrianantoanina (Miami University Ohio) | Some nonlinear problems in the geometry of Banach spaces and their applications ]]<br />
| Tullia Dymarz<br />
|<br />
|-<br />
|Jan 30 '''Wednesday'''<br />
| Talk rescheduled to Feb 15<br />
|<br />
|-<br />
|Jan 31 '''Thursday'''<br />
| Talk rescheduled to Feb 13<br />
|<br />
|-<br />
|Feb 1<br />
| Talk cancelled due to weather<br />
|<br />
| <br />
|<br />
|-<br />
|Feb 5 '''Tuesday, VV 911'''<br />
| [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University)<br />
|[[#Alexei Poltoratski (Texas A&M)| Completeness of exponentials: Beurling-Malliavin and type problems ]]<br />
| Denisov<br />
|<br />
|-<br />
|Feb 6 '''Wednesday, room 911'''<br />
| [https://lc-tsai.github.io/ Li-Cheng Tsai] (Columbia University)<br />
|[[#Li-Cheng Tsai (Columbia University)| When particle systems meet PDEs ]]<br />
| Anderson<br />
|<br />
|-<br />
|Feb 8<br />
| [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)<br />
|[[#Aaron Naber (Northwestern) | A structure theory for spaces with lower Ricci curvature bounds ]]<br />
| Street<br />
|<br />
|-<br />
|Feb 11 '''Monday'''<br />
| [https://www2.bc.edu/david-treumann/materials.html David Treumann] (Boston College)<br />
|[[#David Treumann (Boston College) | Twisting things in topology and symplectic topology by pth powers ]]<br />
| Caldararu<br />
|<br />
|-<br />
| Feb 13 '''Wednesday'''<br />
| [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M)<br />
|[[#Dean Baskin (Texas A&M) | Radiation fields for wave equations ]]<br />
| Street<br />
<br />
|-<br />
| Feb 15 <br />
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)<br />
| [[#Lillian Pierce (Duke University) | Short character sums ]]<br />
| Boston and Street<br />
|<br />
|-<br />
|Feb 22<br />
| [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State)<br />
|[[#Angelica Cueto (The Ohio State University)| Lines on cubic surfaces in the tropics ]]<br />
| Erman and Corey<br />
|<br />
|-<br />
|March 4<br />
| [http://www-users.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) Wasow lecture<br />
|[[# Vladimir Sverak (Minnesota) | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|<br />
|-<br />
|March 8<br />
| [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State)<br />
|[[# TBA| TBA ]]<br />
| Erman<br />
|<br />
|-<br />
|March 15<br />
| Maksym Radziwill (Caltech)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|March 29<br />
| Jennifer Park (OSU)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|April 5<br />
| Ju-Lee Kim (MIT)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 12<br />
| Eviatar Procaccia (TAMU)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 19<br />
| [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)<br />
|[[# TBA| TBA ]]<br />
| Jean-Luc<br />
|<br />
|-<br />
|April 22 '''Monday'''<br />
| [https://justinh.su Justin Hsu] (Madison)<br />
|[[# TBA| TBA ]]<br />
| Lempp<br />
|<br />
|-<br />
|April 26<br />
| [https://www.brown.edu/academics/applied-mathematics/faculty/kavita-ramanan/home Kavita Ramanan] (Brown University)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
|May 3<br />
| Tomasz Przebinda (Oklahoma)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Beata Randrianantoanina (Miami University Ohio)===<br />
<br />
Title: Some nonlinear problems in the geometry of Banach spaces and their applications.<br />
<br />
Abstract: Nonlinear problems in the geometry of Banach spaces have been studied since the inception of the field. In this talk I will outline some of the history, some of modern applications, and some open directions of research. The talk will be accessible to graduate students of any field of mathematics.<br />
<br />
===Lillian Pierce (Duke University)===<br />
<br />
Title: Short character sums <br />
<br />
Abstract: A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.<br />
<br />
===Angelica Cueto (The Ohio State University)===<br />
Title: Lines on cubic surfaces in the tropics<br />
<br />
Abstract: Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-know statement <i>any smooth surface of degree three in P^3 contains exactly 27 lines</i> is known to be false tropically. Work of Vigeland from 2007 provides examples of tropical cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in TP^3.<br />
<br />
In this talk I will explain how to correct this pathology by viewing the surface as a del Pezzo cubic and considering its embedding in P^44 via its anticanonical bundle. The combinatorics of the root system of type E_6 and a tropical notion of convexity will play a central role in the construction. This is joint work in progress with Anand Deopurkar.<br />
<br />
===David Treumann (Boston College)===<br />
<br />
Title: Twisting things in topology and symplectic topology by pth powers<br />
<br />
Abstract: There's an old and popular analogy between circles and finite fields. I'll describe some constructions you can make in Lagrangian Floer theory and in microlocal sheaf theory by taking this analogy extremely literally, the main ingredient is an "F-field." An F-field on a manifold M is a local system of algebraically closed fields of characteristic p. When M is symplectic, maybe an F-field should remind you of a B-field, it can be used to change the Fukaya category in about the same way. On M = S^1 times R^3, this version of the Fukaya category is related to Deligne-Lusztig theory, and I found something like a cluster structure on the Deligne-Lusztig pairing varieties by studying it. On M = S^1 times S^1, Yanki Lekili and I have found that this version of the Fukaya category is related to the equal-characteristic version of the Fargues-Fontaine curve; the relationship is homological mirror symmetry.<br />
<br />
===Dean Baskin (Texas A&M)===<br />
<br />
Title: Radiation fields for wave equations<br />
<br />
Abstract: Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.<br />
<br />
===Jianfeng Lu (Duke University)===<br />
<br />
Title: Density fitting: Analysis, algorithm and applications<br />
<br />
Abstract: Density fitting considers the low-rank approximation of pair products of eigenfunctions of Hamiltonian operators. It is a very useful tool with many applications in electronic structure theory. In this talk, we will discuss estimates of upper bound of the numerical rank of the pair products of eigenfunctions. We will also introduce the interpolative separable density fitting (ISDF) algorithm, which reduces the computational scaling of the low-rank approximation and can be used for efficient algorithms for electronic structure calculations. Based on joint works with Chris Sogge, Stefan Steinerberger, Kyle Thicke, and Lexing Ying.<br />
<br />
===Alexei Poltoratski (Texas A&M)===<br />
<br />
Title: Completeness of exponentials: Beurling-Malliavin and type problems<br />
<br />
Abstract: This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both<br />
problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin<br />
problem was solved in the early 1960s and I will present its classical solution along with modern generalizations<br />
and applications. I will then discuss history and recent progress in the type problem, which stood open for<br />
more than 70 years.<br />
<br />
===Li-Cheng Tsai (Columbia University)===<br />
<br />
Title: When particle systems meet PDEs<br />
<br />
Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems.<br />
<br />
===Aaron Naber (Northwestern)===<br />
<br />
Title: A structure theory for spaces with lower Ricci curvature bounds.<br />
<br />
Abstract: One should view manifolds (M^n,g) with lower Ricci curvature bounds as being those manifolds with a well behaved analysis, a point which can be rigorously stated. It thus becomes a natural question, how well behaved or badly behaved can such spaces be? This is a nonlinear analogue to asking how degenerate can a subharmonic or plurisubharmonic function look like. In this talk we give an essentially sharp answer to this question. The talk will require little background, and our time will be spent on understanding the basic statements and examples. The work discussed is joint with Cheeger, Jiang and with Li.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17026PDE Geometric Analysis seminar2019-02-22T14:57:07Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |TBA ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17025PDE Geometric Analysis seminar2019-02-22T14:56:34Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | PDE aspects of the Navier-Stokes equations and simpler models (Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |TBA ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17024PDE Geometric Analysis seminar2019-02-22T14:56:14Z<p>Ckim: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |TBA ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17023Fall 2019-Spring 20202019-02-22T07:10:06Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17015PDE Geometric Analysis seminar2019-02-21T17:38:34Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |TBA ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17014PDE Geometric Analysis seminar2019-02-21T17:38:12Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |TBA ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17005Fall 2019-Spring 20202019-02-20T18:24:33Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 25<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17004Fall 2019-Spring 20202019-02-20T18:23:29Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 25<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Dec 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17003Fall 2019-Spring 20202019-02-20T18:04:10Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17002Fall 2019-Spring 20202019-02-20T18:02:21Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Date<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Date<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16992PDE Geometric Analysis seminar2019-02-19T12:12:42Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |TBA ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16991PDE Geometric Analysis seminar2019-02-19T03:14:51Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16990PDE Geometric Analysis seminar2019-02-19T03:13:49Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16905PDE Geometric Analysis seminar2019-02-12T15:46:36Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Wenjia Jing===<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16792PDE Geometric Analysis seminar2019-01-31T15:44:01Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16777PDE Geometric Analysis seminar2019-01-29T21:39:54Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16774PDE Geometric Analysis seminar2019-01-29T19:35:34Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February '''PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16716PDE Geometric Analysis seminar2019-01-25T00:22:32Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
| Jan 31 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[# Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16715PDE Geometric Analysis seminar2019-01-25T00:21:19Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
| Jan 31 '''Colloquium 4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[# Dean Baskin | Radiation fields for wave equations]]<br />
| <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16693PDE Geometric Analysis seminar2019-01-23T20:23:00Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16692PDE Geometric Analysis seminar2019-01-23T20:09:10Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25,<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckim