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]]===
===[[Fall 2017 | Tentative schedule for Fall 2017]]===
+
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===
 +
 
 +
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==
  
= PDE GA Seminar Schedule Spring 2017 =
 
  
 
{| cellpadding="8"
 
{| cellpadding="8"
Line 11: 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
 +
|-
 +
|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
 
|-
 
|-
|January 23<br>Special time and location:<br> 3-3:50pm, B325 Van Vleck
+
|Nov 25
| Sigurd Angenent (UW)
+
| Mathew Langford (UT Knoxville)
|[[#Sigurd Angenent | Ancient convex solutions to Mean Curvature Flow]]
+
|[[#Mathew Langford | Concavity of the arrival time ]]
| Kim & Tran
+
| Angenent
 
|-  
 
|-  
 
+
|Dec 9 - Colloquium (4-5PM)
|-
+
| Hui Yu (Columbia)
|January 30
+
|[[#Hui Yu | TBA ]]
| Serguei Denissov (UW)
+
| Tran
|[[#Serguei Denissov | Instability in 2D Euler equation of incompressible inviscid fluid]]
 
| Kim & Tran
 
 
|-  
 
|-  
 
+
|Feb. 3
 
+
| Philippe LeFloch (Sorbonne University and CNRS)
|-
+
|[[#Philippe LeFloch | Nonlinear stability of self-gravitating matter under low decay and weak regularity conditions  ]]
|February 6 - Wasow lecture
+
| Feldman
| Benoit Perthame (University of Paris VI)
 
|[[#| ]]
 
| Jin
 
 
|-  
 
|-  
 +
|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 | 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
 +
|- 
 +
|April 20
 +
| Adrian Tudorascu (WVU)
 +
|[[#Adrian Tudorascu | On the Lagrangian description of the Sticky Particle flow ]]
 +
| Feldman
 +
|- 
 +
|April 27
 +
| Speaker (Institute)
 +
|[[#Speaker | TBA ]]
 +
| Host
 +
|- 
 +
|May 18-21
 +
| Madison Workshop in PDE 2020
 +
|[[#Speaker | TBA ]]
 +
| Tran
 +
|}
  
 +
== Abstracts ==
  
|-
+
===Scott Smith===
|February 13
 
| Bing Wang (UW)
 
|[[#Bing Wang | The extension problem of the mean curvature flow]]
 
| Kim & Tran
 
|-
 
  
|-
+
Title: Recent progress on singular, quasi-linear stochastic PDE
|February 20
 
| Eric Baer (UW)
 
|[[#Eric Baer | Isoperimetric sets inside almost-convex cones]]
 
| Kim & Tran
 
|-  
 
  
|-
+
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.
|February 27
 
| Ben Seeger (University of Chicago)
 
|[[#Ben Seeger | Homogenization of pathwise Hamilton-Jacobi equations ]]
 
| Tran
 
|-
 
  
|-
 
|March 7 - Mathematics Department Distinguished Lecture
 
| Roger Temam (Indiana University) 
 
|[[#Roger Temam | On the mathematical modeling of the humid atmosphere]]
 
| Smith 
 
|-
 
  
 +
===Son Tu===
  
|-
+
Title: State-Constraint static Hamilton-Jacobi equations in nested domains
|March 8 - Analysis/Applied math/PDE seminar
 
| Roger Temam (Indiana University) 
 
|[[#Roger Temam | Weak solutions of the Shigesada-Kawasaki-Teramoto system ]]
 
| Smith
 
|-
 
  
|-
+
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).
|March 13
 
| Sona Akopian (UT-Austin)
 
|[[#Sona Akopian | Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.]]
 
| Kim
 
  
|-
 
|March 27 - Analysis/PDE seminar
 
| Sylvia Serfaty (Courant)
 
|[[#Sylvia Serfaty | Mean-Field Limits for Ginzburg-Landau vortices ]]
 
| Tran
 
  
|-
+
===Jin Woo Jang===
|March 29 - Wasow lecture
 
| Sylvia Serfaty (Courant)
 
|[[#Sylvia Serfaty | Microscopic description of Coulomb-type systems ]]
 
|
 
  
 +
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.
|March 30 <br>Special day (Thursday) and location:<br>  B139 Van Vleck
 
| Gui-Qiang Chen (Oxford)
 
|[[#Gui-Qiang Chen  | Supersonic Flow onto Solid Wedges,
 
Multidimensional Shock Waves and Free Boundary Problems ]]
 
| Feldman
 
  
  
 +
===Stefania Patrizi===
  
|-
+
Title:
|April 3
+
Dislocations dynamics: from microscopic models to macroscopic crystal plasticity
| Zhenfu Wang (Maryland)
 
|[[#Zhenfu Wang | Mean field limit for stochastic particle systems with singular forces]]
 
| Kim
 
  
|-
+
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.
|April 10
 
| Andrei Tarfulea (Chicago)
 
|[[#Andrei Tarfulea | Improved estimates for thermal fluid equations]]
 
| Baer
 
  
|-
 
|April 17
 
| Siao-Hao Guo (Rutgers)
 
|[[# Siao-Hao Guo | Analysis of Velázquez's solution to the mean curvature flow with a type II singularity]]
 
| Lu Wang
 
  
 +
===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.
|April 24
 
| Jianfeng Lu
 
|[[#Jianfeng Lu | TBA]]
 
| Li
 
  
|-
+
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.
|April 25- joint Analysis/PDE seminar
 
| Chris Henderson (Chicago)
 
|[[#Chris Henderson | TBA]]
 
| Lin
 
  
|-
+
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.
|May 1st
 
| Jeffrey Streets (UC-Irvine)
 
|[[#Jeffrey Streets | Generalized Kahler Ricci flow and a generalized Calabi conjecture]]
 
| Bing Wang
 
|}
 
  
=Abstracts=
+
===Albert Ai===
 +
Title: Two dimensional gravity waves at low regularity: Energy estimates
  
===Sigurd Angenent===
+
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.
The Huisken-Hamilton-Gage theorem on compact convex solutions to MCF shows that in forward time all solutions do the same thing, namely, they shrink to a point and become round as they do so. Even though MCF is ill-posed in backward time there do exist solutions that are defined for all t<0 , and one can try to classify all such &ldquo;Ancient Solutions.&rdquo;  In doing so one finds that there is interesting dynamics associated to ancient solutions.  I will discuss what is currently known about these solutions. Some of the talk is based on joint work with Sesum and Daskalopoulos.
 
  
===Serguei Denissov===
+
===Ilyas Khan===
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.
+
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.
  
===Bing Wang===
+
===Mathew Langford===
We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R3. This is a joint work with H.Z. Li.
+
Title: Concavity of the arrival time
  
===Eric Baer===
+
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.
We discuss a recent result showing that a characterization of isoperimetric sets (that is, sets minimizing a relative perimeter functional with respect to a fixed volume constraint) inside convex cones as sections of balls centered at the origin (originally due to P.L. Lions and F. Pacella) remains valid for a class of "almost-convex" cones.  Key tools include compactness arguments and the use of classically known sharp characterizations of lower bounds for the first nonzero Neumann eigenvalue associated to (geodesically) convex domains in the hemisphere.  The work we describe is joint with A. Figalli.
 
  
===Ben Seeger===
+
===Philippe LeFloch===
I present a homogenization result for pathwise Hamilton-Jacobi equations with "rough" multiplicative driving signals. In doing so, I derive a new well-posedness result when the Hamiltonian is smooth, convex, and positively homogenous. I also demonstrate that equations involving multiple driving signals may homogenize or exhibit blow-up.
+
Title: Nonlinear stability of self-gravitating matter under low decay and weak regularity conditions
  
===Sona Akopian===
+
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.  
Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.
 
  
We solve the Cauchy problem associated to an epsilon-parameter family of homogeneous Boltzmann equations for very soft and Coulomb potentials. Proposed in 2013 by Bobylev and Potapenko, the collision kernel that we use is a Dirac mass concentrated at very small angles and relative speeds. The main advantage of such a kernel is that it does not separate its variables (relative speed $u$ and scattering angle $\theta$) and can be viewed as a pseudo-Maxwell molecule collision kernel, which allows for the splitting of the Boltzmann collision operator into its gain and loss terms. Global estimates on the gain term gives us an existence theory for $L^1_k \capL^p$ with any $k\geq 2$ and $p\geq 1.$ Furthermore the bounds we obtain are independent of the epsilon parameter, which allows for analysis of the solutions in the grazing collisions limit, i.e., when epsilon approaches zero and the Boltzmann equation becomes the Landau equation.  
+
(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.  
  
===Sylvia Serfaty===
+
(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.
Mean-Field Limits for Ginzburg-Landau vortices
 
  
Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.
 
  
 +
===Joonhyun La===
 +
Title: On a kinetic model of polymeric fluids
  
===Gui-Qiang Chen===
+
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.
Supersonic Flow onto Solid Wedges, Multidimensional Shock Waves and Free Boundary Problems
 
  
When an upstream steady uniform supersonic flow, governed by the Euler equations,
 
impinges onto a symmetric straight-sided wedge, there are two possible steady oblique shock
 
configurations if the wedge angle is less than the detachment angle -- the steady weak shock
 
with supersonic or subsonic downstream flow (determined by the wedge angle that is less or larger
 
than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which
 
satisfy the entropy conditions.
 
The fundamental issue -- whether one or both of the steady weak and strong shocks are physically
 
admissible solutions -- has been vigorously debated over the past eight decades.
 
In this talk, we discuss some of the most recent developments on the stability analysis
 
of the steady shock solutions in both the steady and dynamic regimes.
 
The corresponding stability problems can be formulated as free boundary problems
 
for nonlinear partial differential equations of mixed elliptic-hyperbolic type, whose
 
solutions are fundamental for multidimensional hyperbolic conservation laws.
 
Some further developments, open problems, and mathematical challenges in this direction
 
are also addressed.
 
  
===Zhenfu Wang===
+
===Yannick Sire===
 +
Title: Minimizers for the thin one-phase free boundary problem
  
Title: Mean field limit for stochastic particle systems with singular forces
+
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.
  
Abstract: We consider large systems of particles interacting through rough interaction kernels. We are able to control the relative entropy between the N-particles distribution
+
===Matthew Schrecker===
and the expected limit which solves the corresponding McKean-Vlasov PDE. This implies the Mean Field limit to the McKean-Vlasov system together with Propagation of Chaos
+
Title: Existence theory and Newtonian limit for 1D relativistic Euler equations
through the strong convergence of all the marginals. The method works at the level of the Liouville equation and relies on precise combinatorics results.
 
  
===Andrei Tarfulea===
+
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.
We consider a model for three-dimensional fluid flow on the torus that also keeps track of the local temperature. The momentum equation is the same as for Navier-Stokes, however the kinematic viscosity grows as a function of the local temperature. The temperature is, in turn, fed by the local dissipation of kinetic energy. Intuitively, this leads to a mechanism whereby turbulent regions increase their local viscosity and
 
dissipate faster. We prove a strong a priori bound (that would fall within the Ladyzhenskaya-Prodi-Serrin criterion for ordinary Navier-Stokes) on the thermally weighted enstrophy for classical solutions to the coupled system.
 
  
===Siao-hao Guo===
+
===Adrian Tudorascu===
Analysis of Velázquez's solution to the mean curvature flow with a type II singularity
+
Title: On the Lagrangian description of the Sticky Particle flow  
  
Velázquez discovered a solution to the mean curvature flow which develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution, the rescaled flow converges in the C^0 sense to a minimal hypersurface which is tangent to Simons' cone at infinity. In this talk, we will present that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we will show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form. This is a joint work with N. Sesum.
+
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 17:16, 12 February 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) 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 Adrian Tudorascu (WVU) On the Lagrangian description of the Sticky Particle flow Feldman
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.

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.

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.)