Analysis Seminar
20212022
Time  Location  Organizers 

Tuesdays at 11am  Zoom (contact organizers for details)  Yuming Paul Zhang and Andrej Zlatoš 
Fall 2021
Date  Speaker  Title + Abstract 

October 5 
Luis Silvestre
University of Chicago 
Regularity estimates for the Boltzmann equation without cutoff
We study the regularization effect of the inhomogeneous Boltzmann equation without cutoff. We obtain a priori estimates for all derivatives of the solution depending only on bounds of its hydrodynamic quantities: mass density, energy density and entropy density. Our analysis applies to the case of moderately soft and hard potentials. We use methods that originated in the study of nonlocal elliptic and parabolic equations: a weak Harnack inequality in the style of De Giorgi, and a Schaudertype estimate. 
October 12 
Yuming Paul Zhang
UCSD 
Homogenization for combustion in random media
We study long time dynamics of combustive processes in random media, modeled by reactiondiffusion equations with random ignition reactions. One expects that under reasonable hypotheses on the randomness, large scale dynamics of solutions to these equations is almost surely governed by a homogeneous HamiltonJacobi equation. While this was previously shown in one dimension as well as for radially symmetric reactions in several dimensions, we prove this phenomenon in the general nonisotropic multidimensional setting. We also show that the rate of convergence of solutions to the HamiltonJacobi dynamics is at least algebraic in the relevant spacetime scales when the initial data is close to an indicator function of a convex set. This talk is based on joint work with Andrej Zlatoš. 
October 19 
Mohandas Pillai
UCSD 
Global, nonscattering solutions to the quintic, focusing semilinear wave equation on $R^{1+3}$
We consider the quintic, focusing semilinear wave equation on $R^{1+3}$, in the radially symmetric setting, and construct infinite time blowup, relaxation, and intermediate types of solutions. More precisely, we first define an admissible class of timedependent length scales, which includes a symbol class of functions. Then, we construct solutions which can be decomposed, for all sufficiently large time, into an AubinTalentini (soliton) solution, rescaled by an admissible length scale, plus radiation (which solves the free 3 dimensional wave equation), plus corrections which decay as time approaches infinity. The solutions include infinite time blowup and relaxation with rates including, but not limited to, positive and negative powers of time, with exponents sufficiently small in absolute value. We also obtain solutions whose soliton component has oscillatory length scales, including ones which converge to zero along one sequence of times approaching infinity, but which diverge to infinity along another such sequence of times. The method of proof is similar to a recent wave maps work of the author, which is itself inspired by matched asymptotic expansions. 
October 26 
Jonas Hirsch
University of Leipzig 
On bounded solutions of linear elliptic operators with measurable coefficients  De Giorgi’s theorem revisited
Abstract 
November 2 
JeanMichel Roquejoffre
University of Toulouse 
Sharp location of the level sets in some reactiondiffusion equations
In a large class of reactiondiffusion equations, the solution starting from a compactly supported initial datum develops a transition between two rest states, that moves at an asymptotically linear rate in time, and whose thickness remains asymptotically bounded in time. The issue is its precise location in time, that is, up to terms that are o(1) as time goes to infinity. This question is well understood in one space dimension; I will discuss what happens in the less well settled multidimensional framework. Joint works with L. Rossi and V. Roussier. 
November 9 
Bjoern Bringmann
IAS 
Invariant Gibbs measures for the threedimensional wave equation with a Hartree nonlinearity
In this talk, we discuss the construction and invariance of the Gibbs measure for a three dimensional wave equation with a Hartreenonlinearity. In the first part of the talk, we construct the Gibbs measure and examine its properties. We discuss the mutual singularity of the Gibbs measure and the socalled Gaussian free field. In contrast, the Gibbs measure for one or twodimensional wave equations is absolutely continuous with respect to the Gaussian free field. In the second part of the talk, we discuss the probabilistic wellposedness of the corresponding nonlinear wave equation, which is needed in the proof of invariance. This was the first theorem proving the invariance of a singular Gibbs measure for any dispersive equation. 
November 23 
Francois Hamel
AixMarseille University 
Symmetry properties for the Euler equations and related semilinear elliptic equations
In this talk, I will discuss radial and onedimensional symmetry properties for the stationary incompressible Euler equations in dimension 2 and some related semilinear elliptic equations. I will show that a steady flow of an ideal incompressible fluid with no stagnation point and tangential boundary conditions in an annulus is a circular flow. The same conclusion holds in complements of disks as well as in punctured disks and in the punctured plane, with some suitable conditions at infinity or at the origin. If possible, I will also discuss the case of parallel flows in twodimensional strips, in the halfplane and in the whole plane. The proofs are based on the study of the geometric properties of the streamlines of the flow and on radial and onedimensional symmetry results for the solutions of some elliptic equations satisfied by the stream function. The talk is based on joint works with N. Nadirashvili. 
November 30 
William Feldman
University of Utah 
Limit shapes of Bernoullitype free boundaries in periodic media
I will discuss some simplified models for the shape of liquid droplets on rough solid surfaces, especially Bernoullitype free boundary problems. In these models small scale roughness leads to large scale nonuniqueness, hysteresis, and anisotropies. In technical terms we need to understand laminating/foliating families of planelike solutions, this is related to ideas of AubryMather theory, but, unlike most results in that area, we need to consider local (but not global) energy minimizers. 
Winter 2022
Date  Speaker  Title + Abstract 

January 11 
SungJin Oh
UC Berkeley 
Blowup and global dynamics for the selfdual ChernSimonsSchrödinger model
The selfdual ChernSimonsSchrödinger model is a gauged cubic NLS on the plane with selfduality, i.e., energy minimizers are given by a firstorder CauchyRiemanntype equation, rather than a secondorder elliptic equation. While this equation shares all formal symmetries with the usual cubic NLS on the plane, the structure of solitary waves is quite different due to selfduality and nonlocality (which stems from the gauge structure). In accordance, this model possesses blowup and global dynamics that are quite different from that of the usual cubic NLS. The goal of this talk is to present some recent results concerning the blowup and global dynamics of this model, with emphasis on a few surprising features of this model such as the impossibility of a ``bubbletree'' blowup and a nonlinear rotational instability of pseudoconformal blowups. This talk is based on joint work with Kihyun Kim (IHES) and Soonsik Kwon (KAIST). 
February 1 
Andrew Lawrie
MIT 
The soliton resolution conjecture for equivariant wave maps
I will present joint work with Jacek Jendrej (CRNS, Sorbonne Paris Nord) on equivariant wave maps with values in the twosphere. We prove that every finite energy equivariant wave map resolves, as time passes, into a superposition of decoupled harmonic maps and radiation, settling the soliton resolution conjecture for this equation. It was proved in works of Côte, and Jia and Kenig, that such a decomposition holds along a sequence of times. We show the resolution holds continuouslyintime via a “noreturn” lemma based on the virial identity. The proof combines a collision analysis of solutions near a multisoliton configuration with concentration compactness techniques. As a byproduct of our analysis we also prove that there are no elastic collisions between pure multisolitons. 
February 8 
Henrik Shahgholian
KTH 
TBA

February 15 
Hung Tran
UW Madison 
TBA

February 22 
Guido De Philippis
NYU 
TBA

March 8 
Javier GomezSerrano
Brown University and University of Barcelona 
TBA

Spring 2022
Date  Speaker  Title + Abstract 

April 12 
Roman Shvydkoy
UIC 
TBA

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