Difference between revisions of "PDE Geometric Analysis seminar"

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The seminar will be held  in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.
+
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]]===
 
===[[Previous PDE/GA seminars]]===
 +
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===
 +
 +
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==
 +
  
== Seminar Schedule Spring 2012 ==
 
 
{| 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)
 +
|- 
 +
|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
 +
|-
 +
|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
 
|-
 
|-
|Feb 6
+
|Nov 25
|Yao Yao (UCLA)
+
| Mathew Langford (UT Knoxville)
|[[#Yao Yao (UCLA)|
+
|[[#Mathew Langford | Concavity of the arrival time ]]
  Degenerate diffusion with nonlocal aggregation: behavior of solutions]]
+
| Angenent
|Kiselev
+
|-
 +
|Dec 9 - Colloquium (4-5PM)
 +
| Hui Yu (Columbia)
 +
|[[#Hui Yu | TBA ]]
 +
| Tran
 +
|-
 +
|Feb. 3
 +
| Philippe LeFloch (Sorbonne University and CNRS)
 +
|[[#Philippe LeFloch | Nonlinear stability of self-gravitating matter under low decay and weak regularity conditions  ]]
 +
| Feldman
 +
|-
 +
|Feb. 10
 +
| Joonhyun La (Stanford)
 +
|[[#Joonhyun La | On a kinetic model of polymeric fluids ]]
 +
| Kim
 +
|-  
 +
|Feb 17
 +
| Yannick Sire (JHU)
 +
|[[#Yannick Sire | Minimizers for the thin one-phase free boundary problem ]]
 +
| Tran
 +
|- 
 +
|Feb 19 - Colloquium (4-5PM)
 +
| Zhenfu Wang (University of Pennsylvania)
 +
|[[#Zhenfu Wang | Quantitative Methods for the Mean Field Limit Problem ]]
 +
| Tran
 +
|- 
 +
|Feb 24
 +
| Matthew Schrecker (UW Madison)
 +
|[[#Matthew Schrecker | Existence theory and Newtonian limit for 1D relativistic Euler equations ]]
 +
| Feldman
 +
|- 
 +
|March 2
 +
| Theodora Bourni (UT Knoxville)
 +
|[[#Speaker | Polygonal Pancakes ]]
 +
| Angenent
 +
|- 
 +
|March 3 -- Analysis seminar
 +
| William Green (Rose-Hulman Institute of Technology)
 +
|[[#William Green  |  Dispersive estimates for the Dirac equation ]]
 +
| Betsy Stovall
 
|-
 
|-
|March 12
+
|March 9
| Xuan Hien Nguyen (Iowa State)
+
| Ian Tice (CMU)
|[[#Xuan Hien Nguyen (Iowa State)|
+
|[[#Ian Tice| Traveling wave solutions to the free boundary Navier-Stokes equations ]]
Gluing constructions for solitons and self-shrinkers under mean curvature flow]]
+
| Kim
|Angenent
+
|- 
|-
+
|March 16
|March 21(Wednesday!), Room 901 Van Vleck
+
| No seminar (spring break)
|Nestor Guillen (UCLA)
+
|[[#Speaker | TBA ]]
|[[#Nestor Guillen (UCLA)|
+
| Host
The local geometry of maps with c-convex potentials]]
+
|-
|Feldman
+
|March 23 (CANCELLED)
|-
+
| Jared Speck (Vanderbilt)
|March 26
+
|[[#Jared Speck | CANCELLED ]]
|Vlad Vicol (University of Chicago)
+
| Schrecker
|[[#Vlad Vicol (U Chicago)|
+
|- 
Shape dependent maximum principles and applications]]
+
|March 30 (CANCELLED)
|Kiselev
+
| Huy Nguyen (Brown)
|-
+
|[[#Huy Nguyen | CANCELLED ]]
|April 9
+
| Kim and Tran
|Charles Smart (MIT)  
+
|-
|[[#Charles Smart (MIT)|
+
|April 6 (CANCELLED, will be rescheduled)
TBA]]
+
| Zhiyan Ding (UW Madison)
|Seeger
+
|[[#Zhiyan Ding | (CANCELLED) Ensemble Kalman Sampling: well-posedness, mean-field limit and convergence analysis ]]
|-
+
| Kim and Tran
|April 16
+
|-
|Jiahong Wu (Oklahoma)
+
|April 13 (CANCELLED)
|[[#Jiahong Wu (Oklahoma State)|
+
| Hyunju Kwon (IAS)
  The 2D Boussinesq equations with partial dissipation]]
+
|[[#Hyunju Kwon | CANCELLED ]]
|Kiselev
+
| Kim
|-
+
|-
|May 14
+
|April 20 (CANCELLED)
|Jacob Glenn-Levin (UT Austin)
+
| Adrian Tudorascu (WVU)
|[[#Jacob Glenn-Levin (UT Austin)|
+
|[[#Adrian Tudorascu | (CANCELLED) On the Lagrangian description of the Sticky Particle flow ]]
TBA]]
+
| Feldman
|Kiselev
+
|
 +
|April 27  
 +
| Christof Sparber (UIC)
 +
|[[#Christof Sparber |  ]]
 +
| Host
 +
|-
 +
|May 18-21
 +
| Madison Workshop in PDE 2020
 +
|[[#Speaker | TBA ]]
 +
| Tran
 
|}
 
|}
  
==Abstracts==
+
== 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.
 +
 
 +
===Philippe LeFloch===
 +
Title: Nonlinear stability of self-gravitating matter under low decay and weak regularity conditions
 +
 
 +
Abstract:  I will present recent progress on the global evolution problem for self-gravitating matter. (1) For Einstein's constraint equations, motivated by a scheme proposed by Carlotto and Schoen I will show the existence of asymptotically Euclidean Einstein spaces with low decay; joint work with T. Nguyen.
  
===Yao Yao (UCLA)===
+
(2) For Einstein's evolution equations in the regime near Minkowski spacetime, I will show the global nonlinear stability of massive matter fields; joint work with Y. Ma.
''Degenerate diffusion with nonlocal aggregation: behavior of solutions''
 
  
The Patlak-Keller-Segel (PKS) equation models the collective motion of
+
(3) For the colliding gravitational wave problem, I will show the existence of weakly regular spacetimes containing geometric singularities across which junction conditions are imposed; joint work with B. Le Floch and G. Veneziano.
cells which are attracted by a self-emitted chemical substance.  While the
 
global well-posedness and finite-time blow up criteria are well known, the
 
asymptotic behaviors of solutions are not completely clear.  In this talk I
 
will present some results on the asymptotic behavior of solutions when
 
there is global existence. The key tools used in the paper are
 
maximum-principle type arguments as well as estimates on mass concentration
 
of solutions. This is a joint work with Inwon Kim.
 
  
===Xuan Hien Nguyen (Iowa State)===
 
  
''Gluing constructions for solitons and self-shrinkers under mean curvature flow''
+
===Joonhyun La===
 +
Title: On a kinetic model of polymeric fluids
  
In the 1990s, Kapouleas and Traizet constructed new examples of minimal surfaces by desingularizing the intersection of existing ones with Scherk surfaces. Using this idea, one can find new examples of self-translating solutions for the mean curvature flow asymptotic at infinity to a finite family of grim reaper cylinders in general position. Recently, it has been shown that it is possible to desingularize the intersection of a sphere and a plane to obtain a family of self-shrinkers under mean curvature flow. I will discuss the main steps and difficulties for these gluing constructions, as well as open problems.
+
Abstract: In this talk, we prove global well-posedness of a system describing behavior of dilute flexible polymeric fluids. This model is based on kinetic theory, and a main difficulty for this system is its multi-scale nature. A new function space, based on moments, is introduced to address this issue, and this function space allows us to deal with larger initial data.
  
===Nestor Guillen (UCLA)===
 
  
We consider the Monge-Kantorovich problem, which consists in
+
===Yannick Sire===
transporting a given measure into another "target" measure in a way
+
Title: Minimizers for the thin one-phase free boundary problem
that minimizes the total cost of moving each unit of mass to its new
 
location. When the transport cost is given by the square of the
 
distance between two points, the optimal map is given by a convex
 
potential which solves the Monge-Ampère equation, in general, the
 
solution is given by what is called a c-convex potential. In recent
 
work with Jun Kitagawa, we prove local Holder estimates of optimal
 
transport maps for more general cost functions satisfying a
 
"synthetic" MTW condition, in particular, the proof does not really
 
use the C^4 assumption made in all previous works. A similar result
 
was recently obtained by Figalli, Kim and McCann using different
 
methods and assuming strict convexity of the target.
 
  
===Charles Smart (MIT)===
+
Abstract: We consider the thin one-phase free boundary problem, associated to minimizing a weighted Dirichlet energy of thefunction in the half-space plus the area of the positivity set of that function restricted to the boundary. I will provide a rather complete picture of the (partial ) regularity of the free boundary, providing content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight related to an anomalous diffusion on the boundary. The approach does not follow the standard one introduced in the seminal work of Alt and Caffarelli. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments which are less reliant on the underlying PDE. This opens several directions of research that I will try to describe.
  
''PDE methods for the Abelian sandpile''
+
===Matthew Schrecker===
 +
Title: Existence theory and Newtonian limit for 1D relativistic Euler equations
  
Abstract: The Abelian sandpile growth model is a deterministic
+
Abstract: I will present the results of my recent work with Gui-Qiang Chen on the Euler equations in the conditions of special relativityI will show how the theory of compensated compactness may be used to obtain the existence of entropy solutions to this system. Moreover, it is expected that as the light speed grows to infinity, solutions to the relativistic Euler equations will converge to their classical (Newtonian) counterparts. The theory we develop is also sufficient to demonstrate this convergence rigorously.
diffusion process for chips placed on the $d$-dimensional integer
 
latticeOne of the most striking features of the sandpile is that it
 
appears to produce terminal configurations converging to a peculiar
 
lattice. One of the most striking features of the sandpile is that it
 
appears to produce terminal configurations converging to a peculiar
 
fractal limit when begun from increasingly large stacks of chips at
 
the origin. This behavior defied explanation for many years until
 
viscosity solution theory offered a new perspective.  This is joint
 
work with Lionel Levine and Wesley Pegden.
 
  
===Vlad Vicol (University of Chicago)===
+
===Theodora Bourni===
 +
Title: Polygonal Pancakes
  
Title: Shape dependent maximum principles and applications
+
Abstract: We study ancient collapsed solutions to mean curvature flow, $\{M^n_t\}_{t\in(-\infty,0)}$, in terms of their squash down: $\Omega_*=\lim_{t\to-\infty}\frac{1}{-t} M_t$. We show that $\Omega_*$ must be a convex body which circumscribes $S^1$ and for every such $\Omega_*$ we construct a solution with this prescribed squash down. Our analysis includes non-compact examples, in which setting we disprove a conjecture of White stating that all eternal solutions must be translators. This is joint work with Langford and Tinaglia.
  
Abstract: We present a non-linear lower bound for the fractional Laplacian, when
+
===Ian Tice===
evaluated at extrema of a function. Applications to the global well-posedness of active
+
Title: Traveling wave solutions to the free boundary Navier-Stokes equations
scalar equations arising in fluid dynamics are discussed. This is joint work with P.
 
Constantin.
 
  
 +
Abstract: Consider a layer of viscous incompressible fluid bounded below
 +
by a flat rigid boundary and above by a moving boundary.  The fluid is
 +
subject to gravity, surface tension, and an external stress that is
 +
stationary when viewed in coordinate system moving at a constant
 +
velocity parallel to the lower boundary.  The latter can model, for
 +
instance, a tube blowing air on the fluid while translating across the
 +
surface.  In this talk we will detail the construction of traveling wave
 +
solutions to this problem, which are themselves stationary in the same
 +
translating coordinate system.  While such traveling wave solutions to
 +
the Euler equations are well-known, to the best of our knowledge this is
 +
the first construction of such solutions with viscosity.  This is joint
 +
work with Giovanni Leoni.
  
===Jiahong Wu (Oklahoma State)===
 
  
"The 2D Boussinesq equations with partial dissipation"
+
===Zhiyan Ding===
 +
Title: Ensemble Kalman Sampling: well-posedness, mean-field limit and convergence analysis
  
The Boussinesq equations concerned here model geophysical flows such
+
Abstract:  Ensemble Kalman Sampling (EKS) is a method to find iid samples from a target distribution. As of today, why the algorithm works and how it converges is mostly unknown. In this talk, I will focus on the continuous version of EKS with linear forward map, a coupled SDE system. I will talk about its well-posedness and justify its mean-filed limit is a Fokker-Planck equation, whose equilibrium state is the target distribution.
as atmospheric fronts and ocean circulations. Mathematically the 2D Boussinesq
 
equations serve as a lower-dimensional model of the 3D hydrodynamics
 
equations. In fact, the 2D Boussinesq equations retain some key features
 
of the 3D Euler and the Navier-Stokes equations such as the vortex stretching
 
mechanism. The global regularity problem on the 2D Boussinesq equations
 
with partial dissipation has attracted considerable attention in the last few years.
 
In this talk we will summarize recent results on various cases of partial dissipation,
 
present the work of Cao and Wu on the 2D Boussinesq equations with vertical
 
dissipation and vertical thermal diffusion, and explain  the work of Chae and Wu on
 
the critical Boussinesq equations with a logarithmically singular velocity.
 
  
===Jacob Glenn-Levin (UT Austin)===
+
===Adrian Tudorascu===
 +
Title: On the Lagrangian description of the Sticky Particle flow
  
TBA
+
Abstract: R. Hynd has recently proved that for absolutely continuous initial velocities the Sticky Particle system admits solutions described by monotone flow maps in Lagrangian coordinates. We present a generalization of this result to general initial velocities and discuss some consequences. (This is based on ongoing work with M. Suder.)

Latest revision as of 09:08, 23 March 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 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 University and CNRS) Nonlinear stability of self-gravitating matter under low decay and weak regularity conditions Feldman
Feb. 10 Joonhyun La (Stanford) On a kinetic model of polymeric fluids Kim
Feb 17 Yannick Sire (JHU) Minimizers for the thin one-phase free boundary problem Tran
Feb 19 - Colloquium (4-5PM) Zhenfu Wang (University of Pennsylvania) Quantitative Methods for the Mean Field Limit Problem Tran
Feb 24 Matthew Schrecker (UW Madison) Existence theory and Newtonian limit for 1D relativistic Euler equations Feldman
March 2 Theodora Bourni (UT Knoxville) Polygonal Pancakes Angenent
March 3 -- Analysis seminar William Green (Rose-Hulman Institute of Technology) Dispersive estimates for the Dirac equation Betsy Stovall
March 9 Ian Tice (CMU) Traveling wave solutions to the free boundary Navier-Stokes equations Kim
March 16 No seminar (spring break) TBA Host
March 23 (CANCELLED) Jared Speck (Vanderbilt) CANCELLED Schrecker
March 30 (CANCELLED) Huy Nguyen (Brown) CANCELLED Kim and Tran
April 6 (CANCELLED, will be rescheduled) Zhiyan Ding (UW Madison) (CANCELLED) Ensemble Kalman Sampling: well-posedness, mean-field limit and convergence analysis Kim and Tran
April 13 (CANCELLED) Hyunju Kwon (IAS) CANCELLED Kim
April 20 (CANCELLED) Adrian Tudorascu (WVU) (CANCELLED) On the Lagrangian description of the Sticky Particle flow Feldman
April 27 Christof Sparber (UIC) 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.

Philippe LeFloch

Title: Nonlinear stability of self-gravitating matter under low decay and weak regularity conditions

Abstract: I will present recent progress on the global evolution problem for self-gravitating matter. (1) For Einstein's constraint equations, motivated by a scheme proposed by Carlotto and Schoen I will show the existence of asymptotically Euclidean Einstein spaces with low decay; joint work with T. Nguyen.

(2) For Einstein's evolution equations in the regime near Minkowski spacetime, I will show the global nonlinear stability of massive matter fields; joint work with Y. Ma.

(3) For the colliding gravitational wave problem, I will show the existence of weakly regular spacetimes containing geometric singularities across which junction conditions are imposed; joint work with B. Le Floch and G. Veneziano.


Joonhyun La

Title: On a kinetic model of polymeric fluids

Abstract: In this talk, we prove global well-posedness of a system describing behavior of dilute flexible polymeric fluids. This model is based on kinetic theory, and a main difficulty for this system is its multi-scale nature. A new function space, based on moments, is introduced to address this issue, and this function space allows us to deal with larger initial data.


Yannick Sire

Title: Minimizers for the thin one-phase free boundary problem

Abstract: We consider the thin one-phase free boundary problem, associated to minimizing a weighted Dirichlet energy of thefunction in the half-space plus the area of the positivity set of that function restricted to the boundary. I will provide a rather complete picture of the (partial ) regularity of the free boundary, providing content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight related to an anomalous diffusion on the boundary. The approach does not follow the standard one introduced in the seminal work of Alt and Caffarelli. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments which are less reliant on the underlying PDE. This opens several directions of research that I will try to describe.

Matthew Schrecker

Title: Existence theory and Newtonian limit for 1D relativistic Euler equations

Abstract: I will present the results of my recent work with Gui-Qiang Chen on the Euler equations in the conditions of special relativity. I will show how the theory of compensated compactness may be used to obtain the existence of entropy solutions to this system. Moreover, it is expected that as the light speed grows to infinity, solutions to the relativistic Euler equations will converge to their classical (Newtonian) counterparts. The theory we develop is also sufficient to demonstrate this convergence rigorously.

Theodora Bourni

Title: Polygonal Pancakes

Abstract: We study ancient collapsed solutions to mean curvature flow, $\{M^n_t\}_{t\in(-\infty,0)}$, in terms of their squash down: $\Omega_*=\lim_{t\to-\infty}\frac{1}{-t} M_t$. We show that $\Omega_*$ must be a convex body which circumscribes $S^1$ and for every such $\Omega_*$ we construct a solution with this prescribed squash down. Our analysis includes non-compact examples, in which setting we disprove a conjecture of White stating that all eternal solutions must be translators. This is joint work with Langford and Tinaglia.

Ian Tice

Title: Traveling wave solutions to the free boundary Navier-Stokes equations

Abstract: Consider a layer of viscous incompressible fluid bounded below by a flat rigid boundary and above by a moving boundary.  The fluid is subject to gravity, surface tension, and an external stress that is stationary when viewed in coordinate system moving at a constant velocity parallel to the lower boundary.  The latter can model, for instance, a tube blowing air on the fluid while translating across the surface.  In this talk we will detail the construction of traveling wave solutions to this problem, which are themselves stationary in the same translating coordinate system.  While such traveling wave solutions to the Euler equations are well-known, to the best of our knowledge this is the first construction of such solutions with viscosity.  This is joint work with Giovanni Leoni.


Zhiyan Ding

Title: Ensemble Kalman Sampling: well-posedness, mean-field limit and convergence analysis

Abstract: Ensemble Kalman Sampling (EKS) is a method to find iid samples from a target distribution. As of today, why the algorithm works and how it converges is mostly unknown. In this talk, I will focus on the continuous version of EKS with linear forward map, a coupled SDE system. I will talk about its well-posedness and justify its mean-filed limit is a Fokker-Planck equation, whose equilibrium state is the target distribution.

Adrian Tudorascu

Title: On the Lagrangian description of the Sticky Particle flow

Abstract: R. Hynd has recently proved that for absolutely continuous initial velocities the Sticky Particle system admits solutions described by monotone flow maps in Lagrangian coordinates. We present a generalization of this result to general initial velocities and discuss some consequences. (This is based on ongoing work with M. Suder.)