Difference between revisions of "PDE Geometric Analysis seminar"

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===[[Previous PDE/GA seminars]]===
 
===[[Previous PDE/GA seminars]]===
===[[Fall 2017 | Tentative schedule for Fall 2017]]===
+
===[[Fall 2021-Spring 2022 | Tentative schedule for Fall 2021-Spring 2022]]===
  
= PDE GA Seminar Schedule Spring 2017 =
 
  
{| cellpadding="8"
 
!style="width:20%" align="left" | date 
 
!align="left" | speaker
 
!align="left" | title
 
!style="width:20%" align="left" | host(s)
 
|-
 
|January 23<br>Special time and location:<br> 3-3:50pm, B325 Van Vleck
 
| Sigurd Angenent (UW)
 
|[[#Sigurd Angenent | Ancient convex solutions to Mean Curvature Flow]]
 
| Kim & Tran
 
|-
 
  
|-
+
== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==
|January 30
+
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. 
| Serguei Denissov (UW)
 
|[[#Serguei Denissov | Instability in 2D Euler equation of incompressible inviscid fluid]]
 
| Kim & Tran
 
|-
 
  
 +
'''Week 1 (9/1/2020-9/5/2020)'''
  
|-
+
1. Paul Rabinowitz - The calculus of variations and phase transition problems.
|February 6 - Wasow lecture
+

https://www.youtube.com/watch?v=vs3rd8RPosA
| Benoit Perthame (University of Paris VI)
 
|[[#| ]]
 
| Jin
 
|-
 
  
 +
2. Frank Merle - On the implosion of a three dimensional compressible fluid.
 +
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be 
  
|-
+
'''Week 2 (9/6/2020-9/12/2020)'''
|February 13
 
| Bing Wang (UW)
 
|[[#Bing Wang | The extension problem of the mean curvature flow]]
 
| Kim & Tran
 
|-
 
  
|-
+
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.
|February 20
+
https://www.youtube.com/watch?v=4ndtUh38AU0
| Eric Baer (UW)
 
|[[#Eric Baer | Isoperimetric sets inside almost-convex cones]]
 
| Kim & Tran
 
|-
 
  
|-
+
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI
|February 27
 
| Ben Seeger (University of Chicago)
 
|[[#Ben Seeger | Homogenization of pathwise Hamilton-Jacobi equations ]]
 
| Tran
 
|-  
 
  
|-
 
|March 7 - Mathematics Department Distinguished Lecture
 
| Roger Temam (Indiana University) 
 
|[[#Roger Temam | On the mathematical modeling of the humid atmosphere]]
 
| Smith 
 
|-
 
  
  
|-
+
'''Week 3 (9/13/2020-9/19/2020)'''
|March 8 - Analysis/Applied math/PDE seminar
 
| Roger Temam (Indiana University)
 
|[[#Roger Temam | Weak solutions of the Shigesada-Kawasaki-Teramoto system ]]
 
| Smith
 
|-
 
  
|-
+
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ
|March 13
 
| Sona Akopian (UT-Austin)
 
|[[#Sona Akopian | Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.]]
 
| Kim
 
  
|-
+
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE
|March 27 - Analysis/PDE seminar
 
| Sylvia Serfaty (Courant)
 
|[[#Sylvia Serfaty | Mean-Field Limits for Ginzburg-Landau vortices ]]
 
| Tran
 
  
|-
 
|March 29 - Wasow lecture
 
| Sylvia Serfaty (Courant)
 
|[[#Sylvia Serfaty | Microscopic description of Coulomb-type systems ]]
 
|
 
  
  
|-
+
'''Week 4 (9/20/2020-9/26/2020)'''
|March 30 <br>Special day (Thursday) and location:<br>  B139 Van Vleck
 
| Gui-Qiang Chen (Oxford)
 
|[[#Gui-Qiang Chen  | Supersonic Flow onto Solid Wedges,
 
Multidimensional Shock Waves and Free Boundary Problems ]]
 
| Feldman
 
  
 +
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be
  
 +
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM
  
|-
 
|April 3
 
| Zhenfu Wang (Maryland)
 
|[[#Zhenfu Wang | Mean field limit for stochastic particle systems with singular forces]]
 
| Kim
 
  
|-
 
|April 10
 
| Andrei Tarfulea (Chicago)
 
|[[#Andrei Tarfulea | Improved estimates for thermal fluid equations]]
 
| Baer
 
  
|-
+
'''Week 5 (9/27/2020-10/03/2020)'''
|April 17 <br>Special time and location:<br> 3-3:50pm, B219 Van Vleck
 
| Siao-Hao Guo (Rutgers)
 
|[[# Siao-Hao Guo | Analysis of Velázquez's solution to the mean curvature flow with a type II singularity]]
 
| Lu Wang
 
  
 +
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo
  
|-
+
2.  Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c
|April 24
 
| Jianfeng Lu (Duke)
 
|[[#Jianfeng Lu | Evolution of crystal surfaces: from mesoscopic to continuum models]]
 
| Li
 
  
|-
 
|April 25- joint Analysis/PDE seminar
 
| Chris Henderson (Chicago)
 
|[[#Chris Henderson | A local-in-time Harnack inequality and applications to reaction-diffusion equations
 
  
]]
+
'''Week 6 (10/04/2020-10/10/2020)'''
| Lin
 
  
|-
+
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E
|May 1st(Special time: 4:00-5:00pm)
 
| Jeffrey Streets (UC-Irvine)
 
|[[#Jeffrey Streets | Generalized Kahler Ricci flow and a generalized Calabi conjecture]]
 
| Bing Wang
 
|}
 
  
=Abstracts=
+
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing
 +
http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html
  
===Sigurd Angenent===
 
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.
 
  
===Serguei Denissov===
+
'''Week 7 (10/11/2020-10/17/2020)'''
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.
 
  
 +
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s
  
===Bing Wang===
+
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg
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.
 
  
===Eric Baer===
 
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.
 
  
===Ben Seeger===
+
'''Week 8 (10/18/2020-10/24/2020)'''
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.
 
  
===Sona Akopian===
+
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg
Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.
 
  
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.  
+
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ
  
===Sylvia Serfaty===
+
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.
Mean-Field Limits for Ginzburg-Landau vortices
 
  
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.
 
  
 +
'''Week 9 (10/25/2020-10/31/2020)'''
  
===Gui-Qiang Chen===
+
1. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764
Supersonic Flow onto Solid Wedges, Multidimensional Shock Waves and Free Boundary Problems
 
  
When an upstream steady uniform supersonic flow, governed by the Euler equations,
+
2.
impinges onto a symmetric straight-sided wedge, there are two possible steady oblique shock
 
configurations if the wedge angle is less than the detachment angle -- the steady weak shock
 
with supersonic or subsonic downstream flow (determined by the wedge angle that is less or larger
 
than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which
 
satisfy the entropy conditions.
 
The fundamental issue -- whether one or both of the steady weak and strong shocks are physically
 
admissible solutions -- has been vigorously debated over the past eight decades.
 
In this talk, we discuss some of the most recent developments on the stability analysis
 
of the steady shock solutions in both the steady and dynamic regimes.
 
The corresponding stability problems can be formulated as free boundary problems
 
for nonlinear partial differential equations of mixed elliptic-hyperbolic type, whose
 
solutions are fundamental for multidimensional hyperbolic conservation laws.
 
Some further developments, open problems, and mathematical challenges in this direction
 
are also addressed.
 
  
===Zhenfu Wang===
+
'''Week 10 (11/1/2020-11/7/2020)'''
  
Title: Mean field limit for stochastic particle systems with singular forces
+
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be
  
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
+
2.  
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
 
through the strong convergence of all the marginals. The method works at the level of the Liouville equation and relies on precise combinatorics results.
 
  
===Andrei Tarfulea===
 
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
 
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.
 
  
===Siao-hao Guo===
+
{| cellpadding="8"
Analysis of Velázquez's solution to the mean curvature flow with a type II singularity
+
!style="width:20%" align="left" | date 
 
+
!align="left" | speaker
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.
+
!align="left" | title
 
+
!style="width:20%" align="left" | host(s)
===Jianfeng Lu===
+
|-  
Evolution of crystal surfaces: from mesoscopic to continuum models
+
|}
 
 
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.
 
 
 
===Chris Henderson===
 
A local-in-time Harnack inequality and applications to reaction-diffusion equations
 
  
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.
+
== Abstracts ==
  
 +
===  ===
  
===Jeffrey Streets===
+
Title: 
Generalized Kahler Ricci flow and a generalized Calabi conjecture
 
  
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.
+
Abstract:

Latest revision as of 14:36, 21 October 2020

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 2021-Spring 2022

PDE GA Seminar Schedule Fall 2020-Spring 2021

Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. 

Week 1 (9/1/2020-9/5/2020)

1. Paul Rabinowitz - The calculus of variations and phase transition problems. 
https://www.youtube.com/watch?v=vs3rd8RPosA

2. Frank Merle - On the implosion of a three dimensional compressible fluid. https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be 

Week 2 (9/6/2020-9/12/2020)

1. Yoshikazu Giga - On large time behavior of growth by birth and spread. https://www.youtube.com/watch?v=4ndtUh38AU0

2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI


Week 3 (9/13/2020-9/19/2020)

1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ

2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE


Week 4 (9/20/2020-9/26/2020)

1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be

2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM


Week 5 (9/27/2020-10/03/2020)

1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo

2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c


Week 6 (10/04/2020-10/10/2020)

1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E

2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html


Week 7 (10/11/2020-10/17/2020)

1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s

2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg


Week 8 (10/18/2020-10/24/2020)

1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg

2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ

Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.


Week 9 (10/25/2020-10/31/2020)

1. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764

2.

Week 10 (11/1/2020-11/7/2020)

1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be

2.


date speaker title host(s)

Abstracts

Title:

Abstract: