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= PDE and Geometric Analysis Seminar - Fall 2010=
+
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]]===
 +
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===
 +
 
 +
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==
  
The seminar will be held  in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.
 
  
== Seminar Schedule ==
 
 
{| 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)
|-
+
|-
|Sept 13
+
|Sep 9
|Fausto Ferrari (Bologna)
+
| Scott Smith (UW Madison)
|[[#Fausto Ferrari (Bologna)|
+
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]
''Semilinear PDEs and some symmetry properties of stable solutions'']]
+
| Kim and Tran
|Feldman
+
|-
|-
+
|Sep 14-15
|Sept 27
+
|  
|Arshak Petrosyan (Purdue)
+
|[[ # |AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html  ]]
|[[#Arshak Petrosyan (Purdue)|
+
|
''Nonuniqueness in a free boundary problem from combustion'']]
+
|-
|Feldman
+
|Sep 23
|-
+
| Son Tu (UW Madison)
|Oct 7, Thursday, 4:30 pm, Room: 901 Van Vleck. '''Special day, time & room.'''
+
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]
|Changyou Wang (U. of Kentucky)
+
| Kim and Tran
|[[#Changyou Wang (U. of Kentucky)|
+
|-
''Phase transition for higher dimensional wells'']]
+
|Sep 28-29, VV901
|Feldman
+
| https://www.ki-net.umd.edu/content/conf?event_id=993
|-
+
|  |  Recent progress in analytical aspects of kinetic equations and related fluid models 
|Oct 11
+
|Philippe LeFloch (Paris VI)
+
|- 
|[[#Philippe LeFloch (Paris VI)|
+
|Oct 7
''Kinetic relations for undercompressive shock waves and propagating phase boundaries'']]
+
| Jin Woo Jang (Postech)
|Feldman
+
|[[#Jin Woo Jang| On a Cauchy problem for the Landau-Boltzmann equation ]]
|-
+
| Kim
|Oct 29 Friday 2:30pm, Room: B115 Van Vleck.    '''Special day, time & room.'''
+
|-
|[http://www.ima.umn.edu/~imitrea/ Irina Mitrea] (IMA)
+
|Oct 14
|[[#Irina Mitrea |
+
| Stefania Patrizi (UT Austin)
''Boundary Value Problems for Higher Order Differential Operators'']]
+
|[[#Stefania Patrizi | Dislocations dynamics: from microscopic models to macroscopic crystal plasticity ]]
|[https://www.math.wisc.edu/~wimaw/ WiMaW]
+
| Tran
|-
+
|-
|-
+
|Oct 21
|Nov 1
+
| Claude Bardos (Université Paris Denis Diderot, France)
|Panagiota Daskalopoulos (Columbia U)
+
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]
|[[#Panagiota Daskalopoulos (Columbia U)|
+
| Li
''Ancient solutions to geometric flows'']]
+
|-
|Feldman
+
|Oct 25-27, VV901
|-
+
| https://www.ki-net.umd.edu/content/conf?event_id=1015
|Nov 8
+
|| Forward and Inverse Problems in Kinetic Theory
|Maria Gualdani (UT Austin)
+
| Li
|[[#Maria Gualdani (UT Austin)|
+
|-  
''A nonlinear diffusion model in mean-field games'']]
+
|Oct 28
|Feldman
+
| Albert Ai (UW Madison)
|-
+
|[[#Albert Ai | Two dimensional gravity waves at low regularity: Energy estimates  ]]
|Nov 18 Thursday 1:20pm  Room: 901 Van Vleck '''Special day & time.'''
+
| Ifrim
|Hiroshi Matano (Tokyo University)
+
|-
|[[#Hiroshi Matano (Tokyo University)|
+
|Nov 4
''Traveling waves in a sawtoothed cylinder and their homogenization limit'']]
+
| Yunbai Cao (UW Madison)
|Angenent & Rabinowitz
+
|[[#Yunbai Cao | Vlasov-Poisson-Boltzmann system in Bounded Domains]]
|-
+
| Kim and Tran
|Nov 29
+
|-  
|Ian Tice (Brown University)
+
|Nov 18
|[[#Ian Tice (Brown University)|
+
| Ilyas Khan (UW Madison)
''TBA'']]
+
|[[#Ilyas Khan | The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension ]]
|Feldman
+
| Kim and Tran
|-
 
|Dec. 8 Wed 2:25pm,  Room: 901 Van Vleck. '''Special day, time & room.'''
 
|Hoai Minh Nguyen (NYU-Courant Institute)
 
|[[#Hoai Minh Nguyen (NYU-Courant Institute)|
 
''TBA'']]
 
|Feldman
 
|-
 
|Date TBA
 
|Mikhail Feldman (UW Madison)
 
|''TBA''
 
|Local speaker
 
 
|-
 
|-
|Date TBA
+
|Nov 25
|Sigurd Angenent (UW Madison)
+
| Mathew Langford (UT Knoxville)
|''TBA''
+
|[[#Mathew Langford | Concavity of the arrival time ]]
|Local speaker
+
| 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
 +
|- 
 +
|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 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
 +
|- 
 +
|April 6 (CANCELLED, will be rescheduled)
 +
| Zhiyan Ding (UW Madison)
 +
|[[#Zhiyan Ding | (CANCELLED) Ensemble Kalman Sampling: well-posedness, mean-field limit and convergence analysis ]]
 +
| 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 ==
 
== Abstracts ==
===Fausto Ferrari (Bologna)===
 
''Semilinear PDEs and some symmetry properties of stable solutions''
 
  
I will deal with stable solutions of semilinear elliptic PDE's
+
===Scott Smith===
and some of their symmetry's properties. Moreover, I will introduce some weighted Poincaré inequalities obtained by combining the notion of stable solution with the definition of weak solution.
+
 
 +
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.
  
===Arshak Petrosyan (Purdue)===
+
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.
''Nonuniqueness in a free boundary problem from combustion''
 
  
We consider a parabolic free boundary problem with a fixed gradient 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.
which serves as a simplified model for the propagation of premixed equidiffusional
 
flames. We give a rigorous justification of an example due to J.L. V ́azquez that  
 
the initial data in the form of two circular humps leads to the nonuniqueness of limit  
 
solutions if the supports of the humps touch at the time of their maximal expansion.
 
  
This is a joint work with Aaron Yip.
+
===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.
  
===Changyou Wang (U. of Kentucky)===
+
===Ilyas Khan===
''Phase transition for higher dimensional wells''
+
Title: The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension.
  
For a potential function <math>F</math> that has two global minimum sets consisting of two compact connected
+
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.
Riemannian submanifolds in <math style="vertical-align=100%" >\mathbb{R}^k</math>, we consider the singular perturbation problem:
 
  
Minimizing <math>\int \left(|\nabla u|^2+\frac{1}{\epsilon^2} F(u)\right)</math> under given Dirichlet boundary data.
+
===Mathew Langford===
 +
Title: Concavity of the arrival time
  
I will discuss a recent joint work with F.H.Lin and X.B.Pan on the asymptotic,  as  the parameter <math>\epsilon</math>
+
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.
tends to zero, in terms of the area of minimal hypersurface interfaces, the minimal connecting energy, and
 
the energy of minimizing harmonic maps into the phase manifolds under both Dirichlet and partially free boundary
 
data. Our results in particular addressed the static case of the so-called Keller-Rubinstein-Sternberg problem.
 
  
===Philippe LeFloch (Paris VI)===
+
===Philippe LeFloch===
''Kinetic relations for undercompressive shock waves and propagating phase boundaries''
+
Title: Nonlinear stability of self-gravitating matter under low decay and weak regularity conditions
  
I will discuss the existence and properties of shock wave solutions to nonlinear hyperbolic systems that are small-scale dependent and, especially, contain undercompressive shock waves or propagating phase boundaries. Regularization-sensitive patterns often arise in continuum physics, especially in complex fluid flows. The so-called kinetic relation is introduced to characterize the correct dynamics of these nonclassical waves, and is tied to a higher-order regularization induced by a more complete model that takes into account additional small-scale physics.  In the present lecture, I will especially explain the techniques of Riemann problems, Glimm-type scheme, and total variation functionals adapted to nonclassical shock waves.  
+
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.
  
===Irina Mitrea===
 
''Boundary Value Problems for Higher Order Differential Operators''
 
  
As is well known, many phenomena in engineering and mathematical physics
+
===Joonhyun La===
can be modeled by means of boundary value problems for a certain elliptic
+
Title: On a kinetic model of polymeric fluids
differential operator L in a domain D.
 
  
When L is a differential operator of second order a variety of tools
+
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.
are available for dealing with such problems including boundary integral
 
methods,
 
variational methods, harmonic measure techniques, and methods based on
 
classical
 
harmonic analysis. The situation when the differential operator has higher order
 
(as is the case for instance with anisotropic plate bending when one
 
deals with
 
fourth order) stands in sharp contrast with this as only fewer options
 
could be
 
successfully implemented. Alberto Calderon, one of the founders of the
 
modern theory
 
of Singular Integral Operators, has advocated in the seventies the use
 
of layer potentials
 
for the treatment of higher order elliptic boundary value problems.
 
While the
 
layer potential method has proved to be tremendously successful in the
 
treatment
 
of second order problems, this approach is insufficiently developed to deal
 
with the intricacies of the theory of higher order operators. In fact,
 
it is largely
 
absent from the literature dealing with such problems.
 
  
In this talk I will discuss recent progress in developing a multiple
 
layer potential
 
approach for the treatment of boundary value problems associated with
 
higher order elliptic differential operators. This is done in a very
 
general class
 
of domains which is in the nature of best possible from the point of
 
view of
 
geometric measure theory.
 
  
 +
===Yannick Sire===
 +
Title: Minimizers for the thin one-phase free boundary problem
  
===Panagiota Daskalopoulos (Columbia U)===
+
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.
''Ancient solutions to geometric flows''
 
  
We will discuss the clasification of ancient solutions to nonlinear geometric flows.
+
===Matthew Schrecker===
It is well known that ancient solutions  appear as blow up limits  at a finite time 
+
Title: Existence theory and Newtonian limit for 1D relativistic Euler equations
singularity of the  flow.
 
Special emphasis will be given to the 2-dimensional Ricci flow.
 
In this case we will show that ancient  compact solution
 
is either the Einstein (trivial)  or one of the King-Rosenau solutions.
 
  
===Maria Gualdani (UT Austin)===
+
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.
''A nonlinear diffusion model in mean-field games''
 
  
We present an overview of mean-field games theory and show
+
===Theodora Bourni===
recent results on a free boundary value problem, which models
+
Title: Polygonal Pancakes
price formation dynamics.
 
In such model, the price is formed through a game among infinite number
 
of agents.
 
Existence and regularity results, as well as linear stability, will be shown.
 
  
===Hiroshi Matano (Tokyo University)===
+
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.
''Traveling waves in a sawtoothed cylinder and their homogenization limit''
 
  
My talk is concerned with a curvature-dependent motion of plane
+
===Ian Tice===
curves in a two-dimensional cylinder with spatially undulating
+
Title: Traveling wave solutions to the free boundary Navier-Stokes equations
boundary.  In other words, the boundary has many bumps and we
 
assume that the bumps are aligned in a spatially recurrent manner.
 
  
The goal is to study how the average speed of the traveling wave
+
Abstract: Consider a layer of viscous incompressible fluid bounded below
depends on the geometry of the domain boundary. More specifically,
+
by a flat rigid boundary and above by a moving boundary.  The fluid is
we consider the homogenization problem as the boundary undulation
+
subject to gravity, surface tension, and an external stress that is
becomes finer and finer, and determine the homogenization limit
+
stationary when viewed in coordinate system moving at a constant
of the average speed and the limit profile of the traveling waves.
+
velocity parallel to the lower boundary.  The latter can model, for
Quite surprisingly, this homogenized speed depends only on the  
+
instance, a tube blowing air on the fluid while translating across the
maximal opening angles of the domain boundary and no other
+
surface.  In this talk we will detail the construction of traveling wave
geometrical features are relevant.  
+
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.
  
Next we consider the special case where the boundary undulation
 
is quasi-periodic with ''m'' independent frequencies.  We show that
 
the rate of convergence to the homogenization limit depends on
 
this number ''m''.
 
  
This is joint work with Bendong Lou and Ken-Ichi Nakamura.
+
===Zhiyan Ding===
 +
Title: Ensemble Kalman Sampling: well-posedness, mean-field limit and convergence analysis
  
===Ian Tice (Brown University)===
+
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
  
===Hoai Minh Nguyen (NYU-Courant Institute)===
+
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.)