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 2018 | Tentative schedule for Spring 2018]]===
+
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===
 +
 
 +
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==
 +
 
  
== PDE GA Seminar Schedule Fall 2017 ==
 
 
{| cellpadding="8"
 
{| cellpadding="8"
 
!style="width:20%" align="left" | date   
 
!style="width:20%" align="left" | date   
Line 10: Line 12:
 
!align="left" | title
 
!align="left" | title
 
!style="width:20%" align="left" | host(s)
 
!style="width:20%" align="left" | host(s)
 +
|- 
 +
|Sep 9
 +
| Scott Smith (UW Madison)
 +
|[[#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
 +
|- 
 +
|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
 
|-  
 
|-  
|September 11
+
|Oct 28
|Mihaela Ifrim (UW)
+
| Albert Ai (UW Madison)
|[[#Mihaela Ifrim|  Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation]]
+
|[[#Albert Ai | Two dimensional gravity waves at low regularity: Energy estimates  ]]
| Kim & Tran
+
| 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
 +
|-
 +
|Nov 25
 +
| Mathew Langford (UT Knoxville)
 +
|[[#Mathew Langford | Concavity of the arrival time ]]
 +
| Angenent
 
|-  
 
|-  
|September 18
+
|Dec 9 - Colloquium (4-5PM)
|Longjie Zhang (University of Tokyo)  
+
| Hui Yu (Columbia)
|[[#Longjie Zhang | On curvature flow with driving force starting as singular initial curve in the plane]]
+
|[[#Hui Yu | TBA ]]
| Angenent
+
| Tran
 
|-  
 
|-  
|September 22,
+
|Feb. 3
VV 9th floor hall, 4:00pm
+
| Philippe LeFloch (Sorbonne Université)
|Jaeyoung Byeon (KAIST)  
+
|[[#Speaker | TBA ]]
|[[#Jaeyoung Byeon| Colloquium: Patterns formation for elliptic systems with large interaction forces]]
+
| Feldman
| Rabinowitz
 
 
|-  
 
|-  
|September 25
+
|Feb. 10
| Tuoc Phan (UTK)
+
| Joonhyun La (Stanford)
|[[#Tuoc Phan Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application]]
+
|[[#Joonhyun La | TBA ]]
 +
| Kim
 +
|-  
 +
|Feb 17
 +
| Yannick Sire (JHU)
 +
|[[#Yannick Sire (JHU) | TBA ]]
 
| Tran
 
| Tran
|-  
+
|-
|September 26,
+
|Feb 24
VV B139 4:00pm
+
| Matthew Schrecker (UW Madison)
| Hiroyoshi Mitake (Hiroshima University)
+
|[[#Matthew Schrecker | TBA ]]
|[[#Hiroyoshi Mitake | Joint Analysis/PDE seminar: Derivation of multi-layered interface system and its application]]
+
| Feldman
| Tran
+
|- 
|-  
+
|March 2
|September 29,
+
| Theodora Bourni (UT Knoxville)
VV901 2:25pm
+
|[[#Speaker | TBA ]]
| Dongnam Ko (CMU & SNU)
+
| Angenent
|[[#Dongnam Ko | a joint seminar with ACMS: On the emergence of local flocking phenomena in Cucker-Smale ensembles ]]
+
|-
| Shi Jin & Kim
+
|March 9
|-  
+
| Ian Tice (CMU)
|October 2
+
|[[#Ian Tice| TBA ]]
| No seminar due to a KI-Net conference
+
| Kim
|
+
|-
|
+
|March 16
|-  
+
| No seminar (spring break)
|October 9
+
|[[#Speaker | TBA ]]
| Sameer Iyer (Brown University)
+
| Host
|[[#Sameer Iyer Global-in-x Steady Prandtl Expansion over a Moving Boundary ]]
+
|-
 +
|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 16
+
|April 20
| Jingrui Cheng (UW)
+
| Speaker (Institute)
|[[#Jingrui Cheng | A 1-D semigeostrophic model with moist convection ]]
+
|[[#Speaker | TBA ]]
| Kim & Tran
+
| Host
|-
+
|-   
|October 23
+
|April 27
| Donghyun Lee (UW)
+
| Speaker (Institute)
|[[#Donghyun Lee | The Vlasov-Poisson-Boltzmann system in bounded domains ]]
+
|[[#Speaker | TBA ]]
| Kim & Tran
+
| Host
|-  
+
|-   
|October 30
+
|May 18-21
| Myoungjean Bae (POSTECH)
+
| Madison Workshop in PDE 2020
|[[#Myoungjean Bae | 3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system ]]
+
|[[#Speaker | TBA ]]
| Feldman
 
|-
 
|November 6
 
| Jingchen Hu (USTC and UW)
 
|[[#Jingchen Hu |  Shock Reflection and Diffraction Problem with Potential Flow Equation ]]
 
| Kim & Tran
 
|-
 
|November 20
 
| Xiaoqin Guo (UW)
 
|[[#Xiaoqin Guo | Quantitative homogenization and Harnack inequality for a degenerate discrete nondivergence form random operator ]]
 
| Kim & Tran
 
|-  
 
|November 27
 
| Ru-Yu Lai (Minnesota)
 
|[[#Ru-Yu Lai | Inverse problems for Maxwell's equations and its application ]]
 
| Li
 
|-  
 
|December 4
 
| Norbert Pozar (Kanazawa University)
 
|[[#Norbert Pozar | Viscosity solutions for the crystalline mean curvature flow ]]
 
 
| Tran
 
| Tran
|-
 
|December 5/6 (Wednesday), Colloquium
 
| Ryan Hynd (U Penn)
 
|[[#Ryan Hynd |  TBD ]]
 
|
 
|-
 
|December 11 (Monday), Colloquium
 
| Connor Mooney (ETH Zurich)
 
|[[#Connor Mooney|  TBD ]]
 
|
 
 
|}
 
|}
  
==Abstracts==
+
== Abstracts ==
 
 
===Mihaela Ifrim===
 
 
 
Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation
 
 
 
Our goal is to take a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation. While this problem is known to be both completely integrable and globally well-posed in $L^2$, much less seems to be known concerning its long time dynamics. We present that for small localized data the solutions have (nearly) dispersive dynamics almost globally in time. An additional objective is to revisit the $L^2$ theory for the Benjamin-Ono equation and provide a simpler, self-contained approach. This is joined work with Daniel Tataru.
 
 
 
===Longjie Zhang===
 
 
 
On curvature flow with driving force starting as singular initial curve in the plane
 
 
 
We consider a family of axisymmetric curves evolving by its mean curvature with driving force in the plane. However, the initial curve is oriented singularly at origin. We investigate this problem by level set method and give some criteria to judge whether the interface evolution is fattening or not. In the end, we can classify the solutions into three categories and provide the asymptotic behavior in each category. Our main tools in this paper are level set method and intersection number principle.
 
 
 
===Jaeyoung Byeon===
 
 
 
Title: Patterns formation for elliptic systems with large interaction forces
 
 
 
Abstract: Nonlinear elliptic systems arising from nonlinear Schroedinger systems have simple looking reaction terms. The corresponding energy for the reaction terms can be expressed as quadratic forms in terms of density functions.  The i, j-th entry of the matrix for the quadratic form represents the interaction force between the components i and j of the system. If the sign of an entry is positive, the force between the two components is attractive; on the other hand, if it is negative, it is repulsive. When the interaction forces between different components are large, the network structure of attraction and repulsion between components might produce several interesting patterns for solutions. As a starting point to study the general pattern formation structure for systems with a large number of components, I will first discuss the simple case of 2-component systems, and then the much more complex case of 3-component systems.
 
 
 
  
===Tuoc Phan===
+
===Scott Smith===
Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application.
 
  
Abstract: In this talk, we first introduce a problem on the existence of global time smooth solutions for a system of cross-diffusion equations. We then recall some classical results on regularity theories, and show that to solve our problem, new results on regularity theory estimates of Calderon-Zygmund type for gradients of solutions to a class of parabolic equations in Lebesgue spaces are required. We then discuss a result on Calderon-Zygmnud type estimate in the concrete setting to solve our
+
Title: Recent progress on singular, quasi-linear stochastic PDE
mentioned problem regarding the system of cross-diffusion equations. The remaining part of the talk will be focused on some new generalized results on regularity gradient estimates for some general class of quasi-linear parabolic equations. Regularity estimates for gradients of solutions in Lorentz spaces will be presented. Ideas of the proofs for the results are given.
 
  
===Hiroyoshi Mitake===
+
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.
Derivation of multi-layered interface system and its application
 
  
Abstract: In this talk, I will propose a multi-layered interface system which can be formally derived by the singular limit of the weakly coupled system of  the Allen-Cahn equation.  By using the level set approach, this system can be written as a quasi-monotone degenerate parabolic system. We give results of the well-posedness of viscosity solutions, and study the singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.
 
  
 +
===Son Tu===
  
===Dongnam Ko===
+
Title: State-Constraint static Hamilton-Jacobi equations in nested domains
On the emergence of local flocking phenomena in Cucker-Smale ensembles
 
  
Emergence of flocking groups are often observed in many complex network systems. The Cucker-Smale model is one of the flocking model, which describes the dynamics of attracting particles. This talk concerns time-asymptotic behaviors of Cucker-Smale particle ensembles, especially for mono-cluster and bi-cluster flockings. The emergence of flocking phenomena is determined by sufficient initial conditions, coupling strength, and communication weight decay. Our asymptotic analysis uses the Lyapunov functional approach and a Lagrangian formulation of the coupled system. We derive a system of differential inequalities for the functionals that measure the local fluctuations and group separations along particle trajectories. The bootstrapping argument is the key idea to prove the gathering and separating behaviors of Cucker-Smale particles simultaneously.
+
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).
  
===Sameer Iyer===
 
Title: Global-in-x Steady Prandtl Expansion over a Moving Boundary.
 
  
Abstract: I will outline the proof that steady, incompressible Navier-Stokes flows posed over the moving boundary, y = 0, can be decomposed into Euler and Prandtl flows globally in the tangential variable, assuming a sufficiently small velocity mismatch. The main obstacles in the analysis center around obtaining sharp decay rates for the linearized profiles and the remainders. The remainders are controlled via a high-order energy method, supplemented with appropriate embedding theorems, which I will present.
+
===Jin Woo Jang===
  
===Jingrui Cheng===
+
Title: On a Cauchy problem for the Landau-Boltzmann equation
  
A 1-D semigeostrophic model with moist convection.
+
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.
  
We consider a simplified 1-D model of semigeostrophic system with moisture, which describes moist convection in a single column in the atmosphere. In general, the solution is non-continuous and it is nontrivial part of the problem to find a suitable definition of weak solutions. We propose a plausible definition of such weak solutions which describes the evolution of the probability distribution of the physical quantities, so that the equations hold in the sense of almost everywhere. Such solutions are constructed from a discrete scheme which obeys the physical principles. This is joint work with Mike Cullen, together with Bin Cheng, John Norbury and Matthew Turner.
 
  
===Donghyun Lee===
+
===Stefania Patrizi===
  
We construct a unique global-in-time solution to the Vlasov-Poisson-Boltzmann system in convex domains with the diffuse boundary condition. Moreover we prove an exponential convergence of distribution function toward the global Maxwellian.
+
Title:
 +
Dislocations dynamics: from microscopic models to macroscopic crystal plasticity
  
===Myoungjean Bae===
+
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.
  
3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system.
 
  
I will present a recent result on the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler–Poisson system in a cylinder supplemented with non-small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl (=angular momentum density) component.  This talk is based on a joint work with S. Weng (Wuhan University, China).
+
===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
  
===Jingchen Hu===
+
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.
  
Shock Reflection and Diffraction Problem with Potential Flow Equation
+
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.
  
In this talk, we will present our work on nonsymmetric shock reflection and diffraction problem, the equation concerned is potential flow equation, which is a simplification of Euler System, mainly based on the assumption that flow has no vortex. We showed in both nonsymmetric reflection case and diffraction case, that physically admissible solution does not exist. This implies that the formation of vortex is essential to maintain the structural stability of shock reflection and diffraction.
+
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.
  
===Xiaoqin Guo===
+
===Albert Ai===
 +
Title: Two dimensional gravity waves at low regularity: Energy estimates
  
Quantitative homogenization and Harnack inequality for a degenerate discrete nondivergence form random operator.
+
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.
  
In the d-dimensional integer lattice $\mathbb Z^d$, $d\ge 2$, we consider a discrete non-divergence form difference operator
+
===Ilyas Khan===
$$
+
Title: The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension.
L_a u(x)=\sum_{i=1}^d a_i(x)[u(x+e_i)+u(x-e_i)-2u(x)]
 
$$
 
where $a(x)=diag(a_1(x),..., a_d(x)), x\in\mathbb Z^d$ are random nonnegative diagonal matrices which are identically distributed and independent and with a positive expectation.
 
A difficulty in studying this problem is that coefficients are allowed to be zero. In this talk, using random walks in random media and its percolative structure, we will present a Harnack inequality and quantitative homogenization result for this random operator. Joint work with N.Berger, M.Cohen and J.-D. Deuschel.
 
  
===Ru-Yu Lai===
+
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.
  
Inverse problems for Maxwell's equations and its application.
+
===Mathew Langford===
 +
Title: Concavity of the arrival time
  
This talk will illustrate the application of complex geometrical optics (CGO) solutions to Maxwell's 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.
First, I will explain the increasing stability behavior of coefficients for Maxwell equations.  
 
In particular, by using CGO solutions, the stability estimate of the conductivity is improving when frequency is growing.
 
Second, I will describe the construction of new families of accelerating and almost nondiffracting beams for Maxwell's equations.
 
They have the form of wave packets that propagate along circular trajectories while almost preserving a trasverse intensity profile.
 

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.