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 2016 | Tentative schedule for Fall 2016]]===
+
===[[Fall 2021-Spring 2022 | Tentative schedule for Fall 2021-Spring 2022]]===
  
= Seminar Schedule Spring 2016 =
 
{| cellpadding="8"
 
!align="left" | date 
 
!align="left" | speaker
 
!align="left" | title
 
!align="left" | host(s)
 
|-
 
|January 25
 
||Tianling Jin (HKUST and Caltech)
 
|[[#Tianling Jin | Holder gradient estimates for parabolic homogeneous p-Laplacian equations  ]]
 
| Zlatos
 
|-
 
|February 1
 
|Russell Schwab (Michigan State University)
 
|[[#Russell Schwab | Neumann homogenization via integro-differential methods  ]]
 
| Lin
 
|-
 
|February 8
 
|Jingrui Cheng (UW Madison)
 
|[[#Jingrui Cheng | Semi-geostrophic system with variable Coriolis parameter ]]
 
| Tran & Kim
 
|-
 
|February 15
 
|Paul Rabinowitz (UW Madison)
 
|[[#Paul Rabinowitz | On A Double Well Potential System  ]]
 
| Tran & Kim
 
|-
 
|February 22
 
|Hong Zhang (Brown)
 
|[[#Hong Zhang | On an elliptic equation arising from composite material ]]
 
| Kim
 
|-
 
|February 29
 
|Aaron Yip (Purdue university)
 
|[[#Aaron Yip |  Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media ]]
 
| Tran
 
|-
 
|March 7
 
|Hiroyoshi Mitake (Hiroshima university)
 
||[[#Hiroyoshi Mitake | Selection problem for fully nonlinear equations]]
 
| Tran
 
|-
 
|March 15
 
|Nestor Guillen (UMass Amherst)
 
|[[#Nestor Guillen | Min-max formulas for integro-differential equations and applications ]]
 
| Lin
 
|-
 
|March 21 (Spring Break)
 
|
 
|[[#  |  ]]
 
|
 
|-
 
|March 28
 
|Ryan Denlinger (Courant Institute)
 
|[[#Ryan Denlinger | The propagation of chaos for a rarefied gas of hard spheres in vacuum ]]
 
| Lee
 
|-
 
|April 4
 
| No seminar
 
||[[#  |  ]]
 
|
 
|-
 
|April 11
 
|Misha Feldman (UW)
 
|[[#Misha Feldman  | Shock reflection, free boundary problems and degenerate elliptic equations ]]
 
|
 
|-
 
|April 14: 2:25 PM in VV 901-Joint with Probability Seminar
 
|Jessica Lin (UW-Madison)
 
|[[#Jessica Lin | Optimal Quantitative Error Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form ]]
 
|-
 
|April 18
 
|Sergey Bolotin (UW)
 
|[[#Sergey Bolotin  | Degenerate billiards in celestial mechanics]]
 
|
 
|-
 
|April 21-24, KI-Net conference: Boundary Value Problems and Multiscale Coupling Methods for Kinetic Equations
 
|Link: http://www.ki-net.umd.edu/content/conf?event_id=493
 
|-
 
|April 25
 
| Moon-Jin Kang (UT-Austin)
 
|[[#  |  ]]
 
| Kim
 
|-
 
|May 3 (Joint Analysis-PDE seminar )
 
|Stanley Snelson (University of Chicago)
 
|[[#  |  ]]
 
| Seeger & Tran.
 
|-
 
|May 16-20, Conference in Harmonic Analysis in Honor of Michael Christ
 
|Link: https://www.math.wisc.edu/ha_2016/
 
|-
 
|}
 
  
=Abstracts=
 
  
===Tianling Jin===
+
== 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. 
  
Holder gradient estimates for parabolic homogeneous p-Laplacian equations
+
'''Week 1 (9/1/2020-9/5/2020)'''
  
We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation
+
1. Paul Rabinowitz - The calculus of variations and phase transition problems.
u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u),
+

https://www.youtube.com/watch?v=vs3rd8RPosA
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.
 
  
===Russell Schwab===
+
2. Frank Merle - On the implosion of a three dimensional compressible fluid.
 +
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be 
  
Neumann homogenization via integro-differential methods
+
'''Week 2 (9/6/2020-9/12/2020)'''
  
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.
+
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.
 +
https://www.youtube.com/watch?v=4ndtUh38AU0
  
===Jingrui Cheng===
+
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI
  
Semi-geostrophic system with variable Coriolis parameter.
 
   
 
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.
 
  
  
===Paul Rabinowitz===
+
'''Week 3 (9/13/2020-9/19/2020)'''
  
On A Double Well Potential System
+
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ
  
We will discuss an elliptic system of partial differential equations of the form
+
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE
\[
 
-\Delta u + V_u(x,u) = 0,\;\;x \in \Omega = \R \times \mathcal{D}\subset \R^n, \;\;\mathcal{D} \; bounded \subset \R^{n-1}
 
\]
 
\[
 
\frac{\partial u}{\partial \nu} = 0 \;\;on \;\;\partial \Omega,
 
\]
 
with $u \in \R^m$,\; $\Omega$ a cylindrical domain in $\R^n$, and $\nu$ the outward pointing normal to $\partial \Omega$.
 
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.
 
When $m=1$, it is a single PDE that arises as an Allen-Cahn model for phase transitions. We will
 
discuss the existence of solutions of \eqref{*} that are heteroclinic from $a^{-}$ to $a^{+}$ or homoclinic to $a^{-}$,
 
i.e. solutions that are of phase transition type.
 
  
This is joint work with Jaeyoung Byeon (KAIST) and Piero Montecchiari (Ancona).
 
  
===Hong Zhang===
 
  
On an elliptic equation arising from composite material
+
'''Week 4 (9/20/2020-9/26/2020)'''
  
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.
+
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
  
===Aaron Yip===
+
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM
  
Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media
 
  
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.
 
  
 +
'''Week 5 (9/27/2020-10/03/2020)'''
  
===Hiroyoshi Mitake===
+
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo
  
Selection problem for fully nonlinear equations
+
2.  Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c
  
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.
 
  
===Nestor Guillen===
+
{| cellpadding="8"
 
+
!style="width:20%" align="left" | date 
Min-max formulas for integro-differential equations and applications
+
!align="left" | speaker
 
+
!align="left" | title
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.
+
!style="width:20%" align="left" | host(s)
 
+
|- 
===Ryan Denlinger===
+
|}
 
 
The propagation of chaos for a rarefied gas of hard spheres in vacuum
 
 
 
We are interested in the rigorous mathematical justification of
 
Boltzmann's equation starting from the deterministic evolution of
 
many-particle systems. O. E. Lanford was able to derive Boltzmann's
 
equation for hard spheres, in the Boltzmann-Grad scaling, on a short
 
time interval. Improvements to the time in Lanford's theorem have so far
 
either relied on a small data hypothesis, or have been restricted to
 
linear regimes. We revisit the small data regime, i.e. a sufficiently
 
dilute gas of hard spheres dispersing into vacuum; this is a regime
 
where strong bounds are available globally in time. Subject to the
 
existence of such bounds, we give a rigorous proof for the propagation
 
of Boltzmann's ``one-sided'' molecular chaos.
 
 
 
===Misha Feldman===
 
 
 
Shock reflection, free boundary problems and degenerate elliptic equations.
 
 
 
Abstract: We will discuss shock reflection problem for compressible gas dynamics, and von Neumann
 
conjectures on transition between regular and Mach reflections. We will discuss existence of solutions
 
of regular reflection structure for potential flow equation, and also regularity of solutions, and
 
properties of the shock curve (free boundary). Our approach is to reduce the shock reflection problem
 
to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type. Open problems
 
will also be discussed, including uniqueness.
 
The talk is based on the joint works with Gui-Qiang Chen, Myoungjean Bae and Wei Xiang.
 
 
 
===Jessica Lin===
 
 
 
Optimal Quantitative Error Estimates in Stochastic
 
Homogenization for Elliptic Equations in Nondivergence Form
 
 
 
Abstract: I will present optimal quantitative error estimates in the
 
stochastic homogenization for uniformly elliptic equations in
 
nondivergence form. From the point of view of probability theory,
 
stochastic homogenization is equivalent to identifying a quenched
 
invariance principle for random walks in a balanced random
 
environment. Under strong independence assumptions on the environment,
 
the main argument relies on establishing an exponential version of the
 
Efron-Stein inequality. As an artifact of the optimal error estimates,
 
we obtain a regularity theory down to microscopic scale, which implies
 
estimates on the local integrability of the invariant measure
 
associated to the process. This talk is based on joint work with Scott
 
Armstrong.
 
 
 
===Sergey Bolotin (UW-Madison)===
 
 
 
Degenerate billiards
 
  
In an ordinary billiard trajectories of a Hamiltonian system
+
== Abstracts ==
are elastically reflected  when colliding with a hypersurface (scatterer).
 
If the scatterer is a submanifold of codimension more than one, then collisions  are rare.
 
Trajectories with infinite number of collisions form a lower dimensional dynamical system.
 
Degenerate billiards appear as limits of ordinary billiards and in celestial mechanics.
 
  
===Moon-Jin Kang===
+
=== ===
  
On contraction of large perturbation of shock waves, and inviscid limit problems
+
Title: 
  
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.
+
Abstract:

Latest revision as of 09:38, 27 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


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


date speaker title host(s)

Abstracts

Title:

Abstract: