Difference between revisions of "PDE Geometric Analysis seminar"

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(PDE GA Seminar Schedule Spring 2018)
(PDE GA Seminar Schedule Fall 2019-Spring 2020)
 
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===[[Previous PDE/GA seminars]]===
 
===[[Previous PDE/GA seminars]]===
===[[Fall 2018 | Tentative schedule for Fall 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 2018-Spring 2019 ==
 
  
  
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!align="left" | title
 
!align="left" | title
 
!style="width:20%" align="left" | host(s)
 
!style="width:20%" align="left" | host(s)
 
 
 
|-   
 
|-   
|September 10,
+
|Sep 9
| Hiroyoshi Mitake (University of Tokyo)
+
| Scott Smith (UW Madison)
|[[#Hiroyoshi Mitake | TBA ]]
+
|[[#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
 +
|-
 +
|Nov 25
 +
| Mathew Langford (UT Knoxville)
 +
|[[#Mathew Langford | Concavity of the arrival time ]]
 +
| Angenent
 +
|-
 +
|Dec 9 - Colloquium (4-5PM)
 +
| Hui Yu (Columbia)
 +
|[[#Hui Yu | TBA ]]
 +
| Tran
 +
|-
 +
|Feb. 3
 +
| Philippe LeFloch (Sorbonne 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
 
| Tran
 
 
|-   
 
|-   
|September 17,
+
|Feb 19 - Colloquium (4-5PM)
| Changyou Wang (Purdue)
+
| Zhenfu Wang (University of Pennsylvania)
|[[#Changyou Wang | TBA ]]
+
|[[#Zhenfu Wang | Quantitative Methods for the Mean Field Limit Problem ]]
 
| Tran
 
| Tran
 
 
|-   
 
|-   
|September 24/26,
+
|Feb 24
| Gunther Uhlmann (UWash)
+
| Matthew Schrecker (UW Madison)
|[[#Gunther Uhlmann | TBA ]]
+
|[[#Matthew Schrecker | Existence theory and Newtonian limit for 1D relativistic Euler equations ]]
| Li
+
| 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 9
 +
| Ian Tice (CMU)
 +
|[[#Ian Tice| Traveling wave solutions to the free boundary Navier-Stokes equations ]]
 +
| Kim
 +
|- 
 +
|March 16
 +
| No seminar (spring break)
 +
|[[#Speaker | TBA ]]
 +
| Host
 +
|- 
 +
|March 23 (CANCELLED)
 +
| Jared Speck (Vanderbilt)
 +
|[[#Jared Speck | CANCELLED ]]
 +
| Schrecker
 +
|- 
 +
|March 30 (CANCELLED)
 +
| Huy Nguyen (Brown)
 +
|[[#Huy Nguyen | CANCELLED ]]
 +
| Kim and Tran
 
|-   
 
|-   
|October 1,
+
|April 6 (CANCELLED, will be rescheduled)
| Matthew Schrecker (UW)
+
| Zhiyan Ding (UW Madison)
|[[#Matthew Schrecker | TBA ]]
+
|[[#Zhiyan Ding | (CANCELLED) Ensemble Kalman Sampling: well-posedness, mean-field limit and convergence analysis ]]
 
| Kim and Tran
 
| Kim and Tran
 +
|- 
 +
|April 13 (CANCELLED)
 +
| Hyunju Kwon (IAS)
 +
|[[#Hyunju Kwon | CANCELLED ]]
 +
| Kim
 +
|- 
 +
|April 20 (CANCELLED)
 +
| Adrian Tudorascu (WVU)
 +
|[[#Adrian Tudorascu | (CANCELLED) On the Lagrangian description of the Sticky Particle flow ]]
 +
| Feldman
 +
|- 
 +
|April 27 
 +
| Christof Sparber (UIC)
 +
|[[#Christof Sparber | (CANCELLED)  ]]
 +
| Host
 +
|- 
 +
|May 18-21
 +
| Madison Workshop in PDE 2020
 +
|[[#Speaker | (CANCELLED) -- Move to 05/2021 ]]
 +
| Tran
 +
|}
  
 +
== Abstracts ==
  
|- 
+
===Scott Smith===
|October 8,
 
| Anna Mazzucato (PSU)
 
|[[#Anna Mazzucato | TBA ]]
 
| Li and Kim
 
 
 
  
|-
+
Title: Recent progress on singular, quasi-linear stochastic PDE
|October 15,
 
| Lei Wu (Lehigh)
 
|[[#Lei Wu | TBA ]]
 
| Kim
 
  
 +
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.
  
|- 
 
|October 22,
 
| Annalaura Stingo (UCD)
 
|[[#Annalaura Stingo | TBA ]]
 
| Mihaela Ifrim
 
  
+
===Son Tu===
|- 
 
|October 29,
 
| Jessica Lin (McGill University)
 
|[[#Jessica Lin | TBA ]]
 
| Tran
 
 
|- 
 
|November 5,
 
| Albert Ai (University of Berkeley)
 
|[[#Albert Ai | TBA ]]
 
| Mihaela Ifrim
 
  
 +
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===
|March 4 2019
 
| Vladimir Sverak (Minnesota)
 
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]
 
| Kim
 
  
== Abstracts ==
+
Title: On a Cauchy problem for the Landau-Boltzmann equation
  
===Dan Knopf===
+
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.
  
Title: Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons
 
  
Abstract: We describe Riemannian (non-Kähler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking Kähler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered by Feldman, Ilmanen, and Knopf in 2003. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-Kähler solutions of Ricci flow that become asymptotically Kähler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of Kähler metrics under Ricci flow.
+
===Stefania Patrizi===
  
===Andreas Seeger===
+
Title:
 +
Dislocations dynamics: from microscopic models to macroscopic crystal plasticity
  
Title: Singular integrals and a problem on mixing flows
+
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.
  
Abstract: The talk will be about  results related to Bressan's mixing problem. We present  an inequality for the change of a  Bianchini semi-norm of characteristic functions under the  flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator  for which one proves bounds  on Hardy spaces. This is joint work with Mahir Hadžić,  Charles Smart and    Brian Street.
 
  
===Sam Krupa===
+
===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
  
Title: Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case
+
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.
  
Abstract: For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (Panov). This single entropy result was proven again by De Lellis, Otto and Westdickenberg in 2004. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In our new work, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi.  Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case. This is joint work with A. Vasseur.
+
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.
  
===Maja Taskovic===
+
===Albert Ai===
 +
Title: Two dimensional gravity waves at low regularity: Energy estimates
  
Title: Exponential tails for the non-cutoff Boltzmann equation
+
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.
  
Abstract: The Boltzmann equation models the motion of a rarefied gas, in which particles interact through binary collisions, by describing the evolution of the particle density function.  The effect of collisions on the density function is modeled by a bilinear integral operator (collision operator) which in many cases has a non-integrable angular kernel.  For a long time the equation was simplified by assuming that this kernel is integrable (the so called Grad's cutoff) with a belief that such an assumption does not affect the equation significantly. However, in the last 20 years it has been observed that a non-integrable singularity carries regularizing properties which motivates further analysis of the equation in this setting.
+
===Ilyas Khan===
 +
Title: The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension.
  
We study behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime by examining the generation and propagation of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. For this purpose we introduce Mittag-Leffler moments which can be understood as a generalization of exponential moments. An interesting aspect of this result is that the singularity rate of the angular kernel affects the order of tails that can be shown to propagate in time. This is based on joint works with Alonso, Gamba, Pavlovic and Gamba, Pavlovic.
+
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
  
===Ashish Kumar Pandey===
+
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.
  
Title: Instabilities in shallow water wave models
+
===Philippe LeFloch===
 +
Title: Nonlinear stability of self-gravitating matter under low decay and weak regularity conditions
  
Abstract: Slow modulations in wave characteristics of a nonlinear, periodic traveling wave in a dispersive medium may develop non-trivial structures which evolve as it propagates. This phenomenon is called modulational instability. In the context of water waves, this phenomenon was observed by Benjamin and Feir and, independently, by Whitham in Stokes' waves. I will discuss a general mechanism to study modulational instability of periodic traveling waves which can be applied to several classes of nonlinear dispersive equations including KdV, BBM, and regularized Boussinesq type equations.
+
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.
  
===Khai Nguyen===
+
(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.
  
Title: Burgers Equation with Some Nonlocal Sources
 
  
Abstract: Consider the Burgers equation with some nonlocal sources, which were derived from models of nonlinear wave with constant frequency.  This talk  will present some recent results on the global existence of entropy weak solutions, priori estimates, and a uniqueness result for both Burgers-Poisson and Burgers-Hilbert equations.  Some open questions will be discussed.
+
===Joonhyun La===
 +
Title: On a kinetic model of polymeric fluids
  
===Hongwei Gao=== 
+
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.
  
Title: Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations
 
  
Abstract: In this talk, we discuss the stochastic homogenization of certain nonconvex Hamilton-Jacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of (level-set) convex Hamiltonians and a sequence of (level-set) concave Hamiltonians through the min-max formula. We provide a monotonicity assumption on the contact values between those stably paired Hamiltonians so as to guarantee the stochastic homogenization. If time permits, we will talk about some homogenization results when the monotonicity assumption breaks down.
+
===Yannick Sire===
 +
Title: Minimizers for the thin one-phase free boundary problem
  
===Huy Nguyen===
+
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.
  
Title : Compressible fluids and active potentials
+
===Matthew Schrecker===
 +
Title: Existence theory and Newtonian limit for 1D relativistic Euler equations
  
Abstract: We consider a class of one dimensional compressible systems with degenerate diffusion coefficients. We establish the fact that the solutions remain smooth as long as the diffusion coefficients do not vanish, and give local and global existence results. The models include the barotropic compressible Navier-Stokes equations, shallow water systems and the lubrication approximation of slender jets. In all these models the momentum equation is forced by the gradient of a solution-dependent potential: the active potential. The method of proof uses the Bresch-Desjardins entropy and the analysis of the evolution of the active potential.
+
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.
  
===In-Jee Jeong===
+
===Theodora Bourni===
 +
Title: Polygonal Pancakes
  
Title: Singularity formation for the 3D axisymmetric Euler equations
+
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 consider the 3D axisymmetric Euler equations on exterior domains  $\{ (x,y,z) : (1 + \epsilon|z|)^2 \le x^2 + y^2  \} $ for any $\epsilon > 0$ so that we can get arbitrarily close to the exterior of a cylinder. We construct a strong local well-posedness class, and show that within this class there exist compactly supported initial data which blows up in finite time. The local well-posedness class consists of velocities which are uniformly Lipschitz in space and have finite energy. Our results were inspired by recent works of Hou-Luo, Kiselev-Sverak, and many others, and the proof builds up on our previous works on 2D Euler and Boussinesq systems. This is joint work with Tarek Elgindi.
+
===Ian Tice===
 +
Title: Traveling wave solutions to the free boundary Navier-Stokes equations
  
===Jeff Calder===
+
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.
  
Title: Nonlinear PDE continuum limits in data science and machine learning
 
  
Abstract: We will present some recent results on PDE continuum limits for (random) discrete problems in data science and machine learning. All of the problems satisfy a type of discrete comparison/maximum principle and so the continuum PDEs are properly interpreted in the viscosity sense. We will present results for nondominated sorting, convex hull peeling, and graph-based semi-supervised learning. Nondominated sorting is an algorithm for arranging points in Euclidean space into layers by repeatedly peeling away coordinatewise minimal points, and the continuum PDE turns out to be a Hamilton-Jacobi equation. Convex hull peeling is used to order data by repeatedly peeling the vertices of the convex hull, and the continuum limit is motion by a power of Gauss curvature. Finally, a recently proposed class of graph-based learning problems have PDE continuum limits corresponding to weighted p-Laplace equations. In each case the continuum PDEs provide insights into the data science/engineering problems, and suggest avenues for fast approximate algorithms based on the PDE interpretations.
+
===Zhiyan Ding===
 +
Title: Ensemble Kalman Sampling: well-posedness, mean-field limit and convergence analysis
  
===Hitoshi Ishii===
+
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.
  
Title: Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory
+
===Adrian Tudorascu===
 +
Title: On the Lagrangian description of the Sticky Particle flow
  
Abstract:   In the lecture, I discuss two asymptotic problems related to Hamilton-Jacobi equations. One concerns the long-time behavior of solutions of time evolutionary Hamilton-Jacobi equations and the other is the so-called vanishing discount problem for stationary Hamilton-Jacobi equations. The last two decades have seen a fundamental importance of weak KAM theory in the asymptotic analysis of Hamilton-Jacobi equations.  I explain briefly the Aubry sets and Mather measures from weak KAM theory and their use in the analysis of the two asymptotic problems above.
+
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 20:41, 6 April 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) (CANCELLED) Host
May 18-21 Madison Workshop in PDE 2020 (CANCELLED) -- Move to 05/2021 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.)