Difference between revisions of "PDE Geometric Analysis seminar"

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===[[Previous PDE/GA seminars]]===
 
===[[Previous PDE/GA seminars]]===
===[[Fall 2016 | Tentative schedule for Spring 2017]]===
+
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===
 +
 
 +
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==
 +
 
  
= PDE GA Seminar Schedule Fall 2016 =
 
 
{| 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 12
+
|Sep 9
| Daniel Spirn (U of Minnesota)
+
| Scott Smith (UW Madison)
|[[#Daniel Spirn Dipole Trajectories in Bose-Einstein Condensates ]]
+
|[[#Scott Smith | 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)
 +
|[[#Son Tu | 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)
 +
|[[#Jin Woo Jang| On a Cauchy problem for the Landau-Boltzmann equation ]]
 
| Kim
 
| Kim
 +
|- 
 +
|Oct 14
 +
| Stefania Patrizi (UT Austin)
 +
|[[#Stefania Patrizi | Dislocations dynamics: from microscopic models to macroscopic crystal plasticity ]]
 +
| Tran
 +
|- 
 +
|Oct 21
 +
| Claude Bardos (Université Paris Denis Diderot, France)
 +
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]
 +
| Li
 +
|- 
 +
|Oct 25-27, VV901
 +
| https://www.ki-net.umd.edu/content/conf?event_id=1015
 +
||  Forward and Inverse Problems in Kinetic Theory
 +
| Li
 +
|-
 +
|Oct 28
 +
| Albert Ai (UW Madison)
 +
|[[#Albert Ai | Two dimensional gravity waves at low regularity: Energy estimates  ]]
 +
| Ifrim
 +
|- 
 +
|Nov 4
 +
| Yunbai Cao (UW Madison)
 +
|[[#Yunbai Cao | Vlasov-Poisson-Boltzmann system in Bounded Domains]]
 +
| Kim and Tran
 +
|- 
 +
|Nov 18
 +
| Ilyas Khan (UW Madison)
 +
|[[#Ilyas Khan | The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension ]]
 +
| Kim and Tran
 
|-
 
|-
|September 19
+
|Nov 25
| Donghyun Lee  (UW-Madison)
+
| Mathew Langford (UT Knoxville)
|[[#Donghyun Lee | The Boltzmann equation with specular boundary condition in convex domains ]]
+
|[[#Mathew Langford | Concavity of the arrival time ]]
 +
| Angenent
 +
|-  
 +
|Dec 9 - Colloquium (4-5PM)
 +
| Hui Yu (Columbia)
 +
|[[#Hui Yu | TBA ]]
 +
| Tran
 +
|-
 +
|Feb. 3
 +
| Philippe LeFloch (Sorbonne Université)
 +
|[[#Speaker | TBA ]]
 
| Feldman
 
| Feldman
|-
+
|-  
|September 26
+
|Feb. 10
| Kevin Zumbrun (Indiana)
+
| Joonhyun La (Stanford)
|[[#Kevin Zumbrun |   A Stable Manifold Theorem for a class of degenerate evolution equations ]]
+
|[[#Joonhyun La | TBA ]]
| Kim  
+
| Kim
|-
+
|-
|October 3
+
|Feb 17
| Will Feldman (UChicago )
+
| Yannick Sire (JHU)
|[[#Will Feldman | Liquid Drops on a Rough Surface  ]]
+
|[[#Yannick Sire (JHU) | TBA ]]
| Lin & Tran
+
| Tran
|-
+
|-
|October 10
+
|Feb 24
| Ryan Hynd (UPenn)
+
| Matthew Schrecker (UW Madison)
|[[#Ryan Hynd | Extremal functions for Morrey’s inequality in convex domains  ]]
+
|[[#Matthew Schrecker | TBA ]]
 
| Feldman
 
| Feldman
|-
+
|-
|October 17
+
|March 2
| Gung-Min Gie (Louisville)
+
| Theodora Bourni (UT Knoxville)
|[[#Gung-Min Gie | Boundary layer analysis of some incompressible flows  ]]
+
|[[#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
 +
| Huy Nguyen (Brown)
 +
|[[#Huy Nguyen | TBA ]]
 +
| organizer
 +
|-
 +
|April 6
 +
| Speaker (Institute)
 +
|[[#Speaker | TBA ]]
 +
| Host
 +
|-
 +
|April 13
 +
| Hyunju Kwon (IAS)
 +
|[[#Hyunju Kwon | TBA ]]
 
| Kim
 
| Kim
|-
+
|-
|October 24
+
|April 20
| Tau Shean Lim (UW Madison)
+
| Speaker (Institute)
|[[#Tau Shean Lim | Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators  ]]
+
|[[#Speaker | TBA ]]
| Kim & Tran
+
| Host
|-
+
|-
|October 31 ('''Special time and room''': B313VV, 3PM-4PM)
+
|April 27
| Tarek Elgindi ( Princeton)
+
| Speaker (Institute)
|[[#Tarek Elgindi | Propagation of Singularities in Incompressible Fluids  ]]
+
|[[#Speaker | TBA ]]
| Lee & Kim
+
| Host
|-
+
|-   
|November 7
+
|May 18-21
| Adrian Tudorascu (West Virginia)
+
| Madison Workshop in PDE 2020
|[[#Adrian Tudorascu | Hamilton-Jacobi equations in the Wasserstein space of probability measures  ]]
+
|[[#Speaker | TBA ]]
| Feldman
+
| Tran
|-
 
|November 14
 
| Alexis Vasseur ( UT-Austin)
 
|[[#Alexis Vasseur | Compressible Navier-Stokes equations with degenerate viscosities  ]]
 
| Feldman
 
|-
 
|November 21
 
| Minh-Binh Tran (UW Madison )
 
|[[#Minh-Binh Tran | Quantum Kinetic Problems  ]]
 
| Hung Tran
 
|-
 
|November 28
 
|  David Kaspar (Brown)
 
|[[#David Kaspar | Kinetics of shock clustering  ]]
 
|Tran
 
|-
 
|December 5 ('''Special time and room''': 3PM-4PM, B313)
 
| Brian Weber (University of Pennsylvania)
 
|[[#Brian Weber | Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds  ]]
 
|Bing Wang
 
|-
 
|December 12
 
|
 
|[[# |    ]]
 
|
 
 
|}
 
|}
  
=Abstracts=
+
== Abstracts ==
 
 
===Daniel Spirn===
 
 
 
Dipole Trajectories in Bose-Einstein Condensates
 
 
 
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.
 
 
 
===Donghyun Lee===
 
 
 
The Boltzmann equation with specular reflection boundary condition in convex domains
 
 
 
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.
 
 
 
===Kevin Zumbrun===
 
 
 
TITLE: A Stable Manifold Theorem for a class of degenerate evolution equations
 
 
 
ABSTRACT:  We establish a Stable Manifold Theorem, with consequent exponential decay to equilibrium, for a class
 
 
 
of degenerate evolution equations $Au'+u=D(u,u)$ with A bounded, self-adjoint, and one-to-one, but not invertible, and
 
 
 
$D$ a bounded, symmetric bilinear map.  This is related to a number of other scenarios investigated recently for which the
 
 
 
associated linearized ODE $Au'+u=0$ is ill-posed with respect to the Cauchy problem.  The particular case studied here
 
 
 
pertains to the steady Boltzmann equation, yielding exponential decay of large-amplitude shock and boundary layers.
 
 
 
 
 
 
 
===Will Feldman===
 
 
 
Liquid Drops on a Rough Surface
 
 
 
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.
 
 
 
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.
 
 
 
===Ryan Hynd===
 
  
Extremal functions for Morrey’s inequality in convex domains
+
===Scott Smith===
  
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.
+
Title: Recent progress on singular, quasi-linear stochastic PDE
  
===Gung-Min Gie ===
+
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.
  
Boundary layer analysis of some incompressible flows
 
 
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.
 
  
===Tau Shean Lim===
+
===Son Tu===
  
Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators
+
Title: State-Constraint static Hamilton-Jacobi equations in nested domains
  
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.
+
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).
  
===Tarek M. ELgindi===
 
  
Propagation of Singularities in Incompressible Fluids
+
===Jin Woo Jang===
  
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.
+
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.
  
===Adrian Tudorascu===
 
  
Hamilton-Jacobi equations in the Wasserstein space of probability measures
+
===Stefania Patrizi===
  
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.
+
Title:
 +
Dislocations dynamics: from microscopic models to macroscopic crystal plasticity
  
===Alexis Vasseur===
+
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.
  
Compressible Navier-Stokes equations with degenerate viscosities 
 
  
We will discuss recent results on the construction of weak solutions for
+
===Claude Bardos===
3D compressible Navier-Stokes equations with degenerate viscosities.
+
Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture
The method is based on the Bresch and Desjardins entropy. The main
 
contribution is to derive  MV type inequalities for the weak solutions,
 
even if it is not verified by the first level of approximation. This
 
provides existence of global solutions in time, for the compressible
 
Navier-Stokes equations,  in three dimensional space, with large initial
 
data, possibly vanishing on the vacuum.
 
  
===Minh-Binh Tran===
+
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.
  
Quantum kinetic problems
+
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.
  
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.
+
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.
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).
 
  
===David Kaspar===
+
===Albert Ai===
 +
Title: Two dimensional gravity waves at low regularity: Energy estimates
  
Kinetics of shock clustering
+
Abstract: In this talk, we will consider the gravity water wave equations in two space dimensions. Our focus is on sharp cubic energy estimates and low regularity solutions. Precisely, we will introduce techniques to prove a new class of energy estimates, which we call balanced cubic estimates. This yields a key improvement over the earlier cubic estimates of Hunter-Ifrim-Tataru, while preserving their scale invariant character and their position-velocity potential holomorphic coordinate formulation. Even without using Strichartz estimates, these results allow us to significantly lower the Sobolev regularity threshold for local well-posedness. This is joint work with Mihaela Ifrim and Daniel Tataru.
  
Suppose we solve a (deterministic) scalar conservation law
+
===Ilyas Khan===
with random initial data.  Can we describe the probability law of the
+
Title: The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension.
solution as a stochastic process in x for fixed later time t?  The
 
answer is yes, for certain Markov initial data, and the probability
 
law factorizes as a product of kernels.  These kernels are obtained by
 
solving a mean-field kinetic equation which most closely resembles the
 
Smoluchowski coagulation equation.  We discuss prior and ongoing work
 
concerning this and related problems.
 
  
===Brian Weber===
+
Abstract: In this talk, we will consider self-shrinking solitons of the mean curvature flow that are smoothly asymptotic to a Riemannian cone in $\mathbb{R}^n$. In 2011, L. Wang proved the uniqueness of self-shrinking ends asymptotic to a cone $C$ in the case of hypersurfaces (codimension 1) by using a backwards uniqueness result for the heat equation due to Escauriaza, Sverak, and Seregin. Later, J. Bernstein proved the same fact using purely elliptic methods. We consider the case of self-shrinkers in high codimension, and outline how to prove the same uniqueness result in this significantly more general case, by using geometric arguments and extending Bernstein’s result.
  
Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds
+
===Mathew Langford===
 +
Title: Concavity of the arrival time
  
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  
+
Abstract:  We present a simple connection between differential Harnack inequalities for hypersurface flows and natural concavity properties of their time-of-arrival functions. We prove these concavity properties directly for a large class of flows by applying a novel concavity maximum principle argument to the corresponding level set flow equations. In particular, this yields a short proof of Hamilton’s differential Harnack inequality for mean curvature flow and, more generally, Andrews’ differential Harnack inequalities for certain “$\alpha$-inverse-concave” flows.
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
 
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.
 

Latest revision as of 09:12, 9 December 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 25-27, VV901 https://www.ki-net.umd.edu/content/conf?event_id=1015 Forward and Inverse Problems in Kinetic Theory Li
Oct 28 Albert Ai (UW Madison) Two dimensional gravity waves at low regularity: Energy estimates Ifrim
Nov 4 Yunbai Cao (UW Madison) Vlasov-Poisson-Boltzmann system in Bounded Domains Kim and Tran
Nov 18 Ilyas Khan (UW Madison) The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension Kim and Tran
Nov 25 Mathew Langford (UT Knoxville) Concavity of the arrival time Angenent
Dec 9 - Colloquium (4-5PM) Hui Yu (Columbia) TBA Tran
Feb. 3 Philippe LeFloch (Sorbonne Université) TBA Feldman
Feb. 10 Joonhyun La (Stanford) TBA Kim
Feb 17 Yannick Sire (JHU) TBA Tran
Feb 24 Matthew Schrecker (UW Madison) TBA Feldman
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 Huy Nguyen (Brown) TBA organizer
April 6 Speaker (Institute) TBA Host
April 13 Hyunju Kwon (IAS) TBA Kim
April 20 Speaker (Institute) TBA Host
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.

Albert Ai

Title: Two dimensional gravity waves at low regularity: Energy estimates

Abstract: In this talk, we will consider the gravity water wave equations in two space dimensions. Our focus is on sharp cubic energy estimates and low regularity solutions. Precisely, we will introduce techniques to prove a new class of energy estimates, which we call balanced cubic estimates. This yields a key improvement over the earlier cubic estimates of Hunter-Ifrim-Tataru, while preserving their scale invariant character and their position-velocity potential holomorphic coordinate formulation. Even without using Strichartz estimates, these results allow us to significantly lower the Sobolev regularity threshold for local well-posedness. This is joint work with Mihaela Ifrim and Daniel Tataru.

Ilyas Khan

Title: The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension.

Abstract: In this talk, we will consider self-shrinking solitons of the mean curvature flow that are smoothly asymptotic to a Riemannian cone in $\mathbb{R}^n$. In 2011, L. Wang proved the uniqueness of self-shrinking ends asymptotic to a cone $C$ in the case of hypersurfaces (codimension 1) by using a backwards uniqueness result for the heat equation due to Escauriaza, Sverak, and Seregin. Later, J. Bernstein proved the same fact using purely elliptic methods. We consider the case of self-shrinkers in high codimension, and outline how to prove the same uniqueness result in this significantly more general case, by using geometric arguments and extending Bernstein’s result.

Mathew Langford

Title: Concavity of the arrival time

Abstract: We present a simple connection between differential Harnack inequalities for hypersurface flows and natural concavity properties of their time-of-arrival functions. We prove these concavity properties directly for a large class of flows by applying a novel concavity maximum principle argument to the corresponding level set flow equations. In particular, this yields a short proof of Hamilton’s differential Harnack inequality for mean curvature flow and, more generally, Andrews’ differential Harnack inequalities for certain “$\alpha$-inverse-concave” flows.