Difference between revisions of "PDE Geometric Analysis seminar"

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(PDE GA Seminar Schedule Fall 2020-Spring 2021)
 
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===[[Previous PDE/GA seminars]]===
 
===[[Previous PDE/GA seminars]]===
===[[Fall 2015 | Tentative schedule for Fall 2015]]===
+
===[[Fall 2021-Spring 2022 | Tentative schedule for Fall 2021-Spring 2022]]===
  
= Seminar Schedule Spring 2015 =
 
{| cellpadding="8"
 
!align="left" | date 
 
!align="left" | speaker
 
!align="left" | title
 
!align="left" | host(s)
 
|-
 
|January 21 (Departmental Colloquium: 4PM, B239)
 
|Jun Kitagawa (Toronto) 
 
|[[#Jun Kitagawa (Toronto)  | Regularity theory for generated Jacobian equations: from optimal transport to geometric optics  ]]
 
|Feldman
 
|-
 
|February 9
 
|Jessica Lin (Madison)
 
|[[#Jessica Lin (Madison)  | Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations ]]
 
|Kim
 
|-
 
|February 17 (Tuesday) (joint with Analysis Seminar: 4PM, B139)
 
|Chanwoo Kim (Madison)
 
|[[#Chanwoo Kim (Madison)  | Hydrodynamic limit from the Boltzmann to the Navier-Stokes-Fourier ]]
 
|Seeger
 
|-
 
|February 23 (special time*, '''3PM, B119''')
 
|  Yaguang Wang (Shanghai Jiao Tong)
 
|[[ #Yaguang Wang | Stability of Three-dimensional Prandtl Boundary Layers ]]
 
|Jin
 
|-
 
|March 2 
 
|Benoit Pausader (Princeton)
 
|[[#Benoit Pausader (Princeton) | Global smooth solutions for the Euler-Maxwell problem for electrons in 2 dimensions]]
 
|Kim
 
|-
 
|March 9
 
|Haozhao Li (University of Science and Technology of China)
 
|[[#Haozhao Li|Regularity scales and convergence of the Calabi flow]]
 
|Wang
 
|-
 
|March 16
 
| Jennifer Beichman (Madison) 
 
|[[#Jennifer Beichman (Madison)  |Nonstandard dispersive estimates and linearized water waves  ]]
 
|  Kim
 
|-
 
|March 23
 
| Ben Fehrman (University of Chicago)
 
|[[#Ben Fehrman (University of Chicago  | On The Existence of an Invariant Measure for Isotropic Diffusions in Random Environments ]]
 
| Lin
 
|-
 
|March 30
 
| Spring recess Mar 28-Apr 5 (S-N)
 
|[[#  |  ]]
 
|
 
|-
 
|April 6
 
| Vera Hur (UIUC)
 
|[[# Vera Hur (UIUC) | Instabilities in nonlinear dispersive waves ]]
 
| Yao
 
|-
 
|April 13
 
| Sung-Jin Oh (Berkeley)
 
|[[#  |  ]]
 
| Kim
 
|-
 
|April 20
 
|Yuan Lou (Ohio State)
 
|[[#Yuan Lou (Ohio State) | TBA]]
 
|Zlatos
 
|-
 
|April 27
 
|
 
|[[#  |  ]]
 
|
 
|-
 
|April 28 (a joint seminar with analysis, 4:00 p.m B139)
 
| Diego Córdoba (ICMAT, Madrid)
 
|[[#  |  ]]
 
| Zlatos
 
|-
 
|May 4
 
|
 
|[[#  |  ]]
 
|
 
|-
 
|}
 
  
  
== Abstracts ==
+
== 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. 
  
===Jun Kitagawa (Toronto)===
+
'''Week 1 (9/1/2020-9/5/2020)'''
  
Regularity theory for generated Jacobian equations: from optimal transport to geometric optics
+
1. Paul Rabinowitz - The calculus of variations and phase transition problems.
 +

https://www.youtube.com/watch?v=vs3rd8RPosA
  
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.
+
2. Frank Merle - On the implosion of a three dimensional compressible fluid.
 +
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be 
  
===Jessica Lin (Madison)===
+
'''Week 2 (9/6/2020-9/12/2020)'''
  
Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations
+
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.
 +
https://www.youtube.com/watch?v=4ndtUh38AU0
  
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.
+
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI
  
  
===Yaguang Wang (Shanghai Jiao Tong)===
 
  
Stability of Three-dimensional Prandtl Boundary Layers
+
'''Week 3 (9/13/2020-9/19/2020)'''
  
In this talk, we shall study the stability of the Prandtl boundary layer
+
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ
equations in three space variables. First, we obtain a well-posedness
 
result of the three-dimensional Prandtl equations under some constraint on
 
its flow structure. It reveals that the classical Burgers equation plays an
 
important role in determining this type of flow with special structure,
 
that avoids the appearance of the complicated secondary flow in the
 
three-dimensional Prandtl boundary layers. Second, we give an instability
 
criterion for the Prandtl equations in three space variables. Both of
 
linear and nonlinear stability are considered. This criterion shows that
 
the monotonic shear flow is linearly stable for the three dimensional
 
Prandtl equations if and only if the tangential velocity field direction is
 
invariant with respect to the normal variable, which is an exact complement
 
to the above well-posedness result for a special flow. This is a joint work
 
with Chengjie Liu and Tong Yang.
 
  
 +
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE
  
===Benoit Pausader (Princeton)===
 
  
Global smooth solutions for the Euler-Maxwell problem for electrons in 2 dimensions
 
  
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.
+
'''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
  
===Haozhao Li (University of Science and Technology of China)===
+
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM
  
Regularity scales and convergence of the Calabi flow
 
  
We define regularity scales to study the behavior of the Calabi flow.
+
{| cellpadding="8"
Based on estimates of the regularity scales, we obtain convergence theorems
+
!style="width:20%" align="left" | date 
of the Calabi flow on extremal K\"ahler surfaces, under the assumption of global existence
+
!align="left" | speaker
of the Calabi flow solutions. Our results partially confirm Donaldson’s conjectural picture for
+
!align="left" | title
the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra
+
!style="width:20%" align="left" | host(s)
assumption that the scalar curvature is uniformly bounded.
+
|- 
 +
|}
  
 +
== Abstracts ==
  
===Jennifer Beichman (UW-Madison)===
+
===  ===
 
 
Nonstandard dispersive estimates and linearized water waves
 
 
 
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.
 
 
 
===Ben Fehrman (University of Chicago)===
 
 
 
On The Existence of an Invariant Measure for Isotropic Diffusions in Random Environments
 
 
 
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.
 
 
 
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.
 
 
 
===Vera Hur===
 
  
Instabilities in nonlinear dispersive waves
+
Title: 
  
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.
+
Abstract:

Latest revision as of 11:49, 20 September 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


date speaker title host(s)

Abstracts

Title:

Abstract: