Difference between revisions of "PDE Geometric Analysis seminar"

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(PDE GA Seminar Schedule Fall 2019-Spring 2020)
 
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===[[Previous PDE/GA seminars]]===
 
===[[Previous PDE/GA seminars]]===
===[[Spring 2015 | Tentative schedule for Spring 2015]]===
+
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===
 +
 
 +
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==
 +
 
  
= Seminar Schedule Fall 2014 =
 
 
{| cellpadding="8"
 
{| cellpadding="8"
!align="left" | date   
+
!style="width:20%" align="left" | date   
!align="left" | speaker                  
+
!align="left" | speaker
 
!align="left" | title
 
!align="left" | title
!align="left" | host(s)
+
!style="width:20%" align="left" | host(s)
|-
+
|-
|September 15
+
|Sep 9
|Greg Kuperberg (UC-Davis)
+
| Scott Smith (UW Madison)
|[[#Greg Kuperberg| ''Cartan-Hadamard and the Little Prince'' ]]
+
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]
|Viaclovsky
+
| Kim and Tran
|-
+
|-
|-
+
|Sep 14-15
|September 22 (joint with Analysis Seminar)
+
|  
|Steve Hofmann (U. of Missouri)
+
|[[ # |AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html  ]]
|[[#Steve Hofmann| ''Quantitative Rectifiability and Elliptic Equations'']]
+
|
|Seeger
+
|-
|-
+
|Sep 23
|-
+
| Son Tu (UW Madison)
|Oct 6th,
+
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]
|Xiangwen Zhang (Columbia University)
+
| Kim and Tran
|[[#Xiangwen Zhang |Alexandrov's Uniqueness Theorem for Convex Surfaces ]]
+
|-
|B.Wang
+
|Sep 28-29, VV901
|-
+
| https://www.ki-net.umd.edu/content/conf?event_id=993
|-
+
|   | Recent progress in analytical aspects of kinetic equations and related fluid models 
|October 13
+
|
|Xuwen Chen (Brown University)[http://www.math.brown.edu/~chenxuwen/]
+
|-
|[[#Xuwen Chen| The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-body Evolution]]
+
|Oct 7
|C.Kim
+
| Jin Woo Jang (Postech)
|-
+
|[[#Jin Woo Jang| On a Cauchy problem for the Landau-Boltzmann equation ]]
|-
+
| Kim
|October 20
+
|-
|Kyudong Choi (UW-Madison)
+
|Oct 14
|[[#Kyudong Choi| Finite time blow up for 1D models for the 3D Axisymmetric Euler Equations the 2D Boussinesq system]]
+
| Stefania Patrizi (UT Austin)
|C.Kim
+
|[[#Stefania Patrizi | Dislocations dynamics: from microscopic models to macroscopic crystal plasticity ]]
|-
+
| Tran
|-
+
|-
|October 27
+
|Oct 21
|Chanwoo Kim (UW-Madison)
+
| Claude Bardos (Université Paris Denis Diderot, France)
|[[#Chanwoo Kim|
+
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]
BV-Regularity of the Boltzmann Equation in Non-Convex Domains]]
+
| Li
|Local
+
|-
|-
+
|Oct 28
|-
+
| Albert Ai (UW Madison)
|November 3
+
|[[#Albert Ai | TBA ]]
|Myoungjean Bae (POSTECH)
+
| Ifrim
|[[#Myoungjean Bae| Recent progress on study of Euler-Poisson system ]]
+
|-
|M.Feldman
+
|Nov 4
|-
+
| Yunbai Cao (UW Madison)
|-
+
|[[#Yunbai Cao | TBA ]]
|November 10
+
| Kim and Tran
|Philip Isett (MIT)
+
|-
|[[#Philip Isett| Hölder Continuous Euler Flows ]]
+
|Nov 11
|C.Kim
+
| Speaker (Institute)
|-
+
|[[#Speaker | TBA ]]
|-
+
| Host
|November 17
+
|-
|Lei Wu
+
|Nov 18
|[[#Lei Wu | Geometric Correction for Diffusive Expansion in Neutron Transport Equation  ]]
+
| Speaker (Institute)
|C.Kim
+
|[[#Speaker | TBA ]]
|-
+
| Host
|-
 
|November 24
 
|Hongnian Huang (Univeristy of New Mexico)
 
|[[#Hongnian Huang| TBA ]]
 
|B.Wang
 
|-
 
|-
 
|December 1
 
|[http://orion.math.iastate.edu/xhnguyen Xuan Hien Nguyen] (Iowa State University)
 
|[[#Xuan Hien Nguyen| TBA ]]
 
|Angenent
 
 
|-
 
|-
 +
|Nov 25
 +
| Mathew Langford (UT Knoxville)
 +
|[[#Speaker | TBA ]]
 +
| Angenent
 +
|- 
 +
|- 
 +
|Feb 17
 +
| Yannick Sire (JHU)
 +
|[[#Yannick Sire (JHU) | TBA ]]
 +
| Tran
 +
|- 
 +
|Feb 24
 +
| Speaker (Institute)
 +
|[[#Speaker | TBA ]]
 +
| Host
 +
|- 
 +
|March 2
 +
| Theodora Bourni (UT Knoxville)
 +
|[[#Speaker | TBA ]]
 +
| Angenent
 +
|- 
 +
|March 9
 +
| Ian Tice (CMU)
 +
|[[#Ian Tice| TBA ]]
 +
| Kim
 +
|- 
 +
|March 16
 +
| No seminar (spring break)
 +
|[[#Speaker | TBA ]]
 +
| Host
 +
|- 
 +
|March 23
 +
| Jared Speck (Vanderbilt)
 +
|[[#Jared Speck | TBA ]]
 +
| SCHRECKER
 +
|- 
 +
|March 30
 +
| Speaker (Institute)
 +
|[[#Speaker | TBA ]]
 +
| Host
 +
|- 
 +
|April 6
 +
| Speaker (Institute)
 +
|[[#Speaker | TBA ]]
 +
| Host
 +
|- 
 +
|April 13
 +
| Speaker (Institute)
 +
|[[#Speaker | TBA ]]
 +
| Host
 +
|- 
 +
|April 20
 +
| Hyunju Kwon (IAS)
 +
|[[#Hyunju Kwon | TBA ]]
 +
| Kim
 +
|- 
 +
|April 27
 +
| Speaker (Institute)
 +
|[[#Speaker | TBA ]]
 +
| Host
 +
|- 
 +
|May 18-21
 +
| Madison Workshop in PDE 2020
 +
|[[#Speaker | TBA ]]
 +
| Tran
 
|}
 
|}
  
== Fall Abstracts ==
+
== Abstracts ==
 +
 
 +
===Scott Smith===
 +
 
 +
Title: Recent progress on singular, quasi-linear stochastic PDE
 +
 
 +
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.
 +
 
  
===Greg Kuperberg===
+
===Son Tu===
''Cartan-Hadamard and the Little Prince.''
 
  
===Steve Hofmann===
+
Title: State-Constraint static Hamilton-Jacobi equations in nested domains
''Quantitative Rectifiability and Elliptic Equations''
 
  
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.
+
Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).
  
===Xiangwen Zhang===
 
''Alexandrov's Uniqueness Theorem for Convex Surfaces''
 
  
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.
+
===Jin Woo Jang===
If time permits, we will also briefly introduce the idea of our recent work on Alexandrov’s theorems for codimension two submanifolds in spacetimes.
 
  
===Xuwen Chen===
+
Title: On a Cauchy problem for the Landau-Boltzmann equation
''The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-body Evolution''
 
  
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.
+
Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.
  
===Kyudong Choi===
 
''Finite time blow up for 1D models for the 3D Axisymmetric Euler Equations the
 
2D Boussinesq system''
 
  
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.
+
===Stefania Patrizi===
  
===Myoungjean Bae===
+
Title:
''Recent progress on study of Euler-Poisson system''
+
Dislocations dynamics: from microscopic models to macroscopic crystal plasticity
  
In this talk, I will present recent progress on the following subjects:
+
Abstract: Dislocation theory aims at explaining the plastic behavior of materials by the motion of line defects in crystals. Peierls-Nabarro models consist in approximating the geometric motion of these defects by nonlocal reaction-diffusion equations. We study the asymptotic  limit of  solutions of  Peierls-Nabarro equations. Different scalings lead to different models at microscopic, mesoscopic and macroscopic scale. This is joint work with E. Valdinoci.
(1) Smooth transonic flow of Euler-Poisson system;
 
(2) Transonic shock of Euler-Poisson system.
 
This talk is based on collaboration with Ben Duan, Chujing Xie and Jingjing Xiao
 
  
===Philip Isett===
 
"Hölder Continuous Euler Flows"
 
  
Motivated by the theory of hydrodynamic turbulence, L. Onsager conjectured
+
===Claude Bardos===
in 1949 that solutions to the incompressible Euler equations with Holder
+
Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture
regularity less than 1/3 may fail to conserve energy.  C. De Lellis and L.
 
Székelyhidi, Jr. have pioneered an approach to constructing such irregular
 
flows based on an iteration scheme known as convex integration.  This
 
approach involves correcting “approximate solutions" by adding rapid
 
oscillations which are designed to reduce the error term in solving the
 
equation.  In this talk, I will discuss an improved convex integration
 
framework, which yields solutions with Holder regularity 1/5-, as well as
 
other related results.
 
  
===Lei Wu===
+
Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.
''Geometric Correction for Diffusive Expansion in Neutron Transport Equation''
 
  
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.
+
Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.
  
===Xuan Hien Nguyen===
+
Eventually the above results are compared with  several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.
"TBA"
 

Latest revision as of 22:31, 13 October 2019

The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

Previous PDE/GA seminars

Tentative schedule for Fall 2020-Spring 2021

PDE GA Seminar Schedule Fall 2019-Spring 2020

date speaker title host(s)
Sep 9 Scott Smith (UW Madison) Recent progress on singular, quasi-linear stochastic PDE Kim and Tran
Sep 14-15 AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html
Sep 23 Son Tu (UW Madison) State-Constraint static Hamilton-Jacobi equations in nested domains Kim and Tran
Sep 28-29, VV901 https://www.ki-net.umd.edu/content/conf?event_id=993 Recent progress in analytical aspects of kinetic equations and related fluid models
Oct 7 Jin Woo Jang (Postech) On a Cauchy problem for the Landau-Boltzmann equation Kim
Oct 14 Stefania Patrizi (UT Austin) Dislocations dynamics: from microscopic models to macroscopic crystal plasticity Tran
Oct 21 Claude Bardos (Université Paris Denis Diderot, France) From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture Li
Oct 28 Albert Ai (UW Madison) TBA Ifrim
Nov 4 Yunbai Cao (UW Madison) TBA Kim and Tran
Nov 11 Speaker (Institute) TBA Host
Nov 18 Speaker (Institute) TBA Host
Nov 25 Mathew Langford (UT Knoxville) TBA Angenent
Feb 17 Yannick Sire (JHU) TBA Tran
Feb 24 Speaker (Institute) TBA Host
March 2 Theodora Bourni (UT Knoxville) TBA Angenent
March 9 Ian Tice (CMU) TBA Kim
March 16 No seminar (spring break) TBA Host
March 23 Jared Speck (Vanderbilt) TBA SCHRECKER
March 30 Speaker (Institute) TBA Host
April 6 Speaker (Institute) TBA Host
April 13 Speaker (Institute) TBA Host
April 20 Hyunju Kwon (IAS) TBA Kim
April 27 Speaker (Institute) TBA Host
May 18-21 Madison Workshop in PDE 2020 TBA Tran

Abstracts

Scott Smith

Title: Recent progress on singular, quasi-linear stochastic PDE

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.


Son Tu

Title: State-Constraint static Hamilton-Jacobi equations in nested domains

Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).


Jin Woo Jang

Title: On a Cauchy problem for the Landau-Boltzmann equation

Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.


Stefania Patrizi

Title: Dislocations dynamics: from microscopic models to macroscopic crystal plasticity

Abstract: Dislocation theory aims at explaining the plastic behavior of materials by the motion of line defects in crystals. Peierls-Nabarro models consist in approximating the geometric motion of these defects by nonlocal reaction-diffusion equations. We study the asymptotic limit of solutions of Peierls-Nabarro equations. Different scalings lead to different models at microscopic, mesoscopic and macroscopic scale. This is joint work with E. Valdinoci.


Claude Bardos

Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture

Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.

Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.

Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.