# Analysis Seminar

## 2021-2022

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 Schauder-type 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 reaction-diffusion 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 Hamilton-Jacobi 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 non-isotropic multidimensional setting. We also show that the rate of convergence of solutions to the Hamilton-Jacobi dynamics is at least algebraic in the relevant space-time 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, non-scattering 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 blow-up, relaxation, and intermediate types of solutions. More precisely, we first define an admissible class of time-dependent length scales, which includes a symbol class of functions. Then, we construct solutions which can be decomposed, for all sufficiently large time, into an Aubin-Talentini (soliton) solution, re-scaled 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 blow-up 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 Jean-Michel Roquejoffre
University of Toulouse
Sharp location of the level sets in some reaction-diffusion equations
In a large class of reaction-diffusion 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 multi-dimensional framework. Joint works with L. Rossi and V. Roussier.
November 9 Bjoern Bringmann
IAS
Invariant Gibbs measures for the three-dimensional 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 Hartree-nonlinearity. 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 so-called Gaussian free field. In contrast, the Gibbs measure for one or two-dimensional wave equations is absolutely continuous with respect to the Gaussian free field. In the second part of the talk, we discuss the probabilistic well-posedness 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
Aix-Marseille University
Symmetry properties for the Euler equations and related semilinear elliptic equations
In this talk, I will discuss radial and one-dimensional 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 two-dimensional strips, in the half-plane 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 one-dimensional 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 Bernoulli-type free boundaries in periodic media
I will discuss some simplified models for the shape of liquid droplets on rough solid surfaces, especially Bernoulli-type free boundary problems. In these models small scale roughness leads to large scale non-uniqueness, hysteresis, and anisotropies. In technical terms we need to understand laminating/foliating families of plane-like solutions, this is related to ideas of Aubry-Mather 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 Sung-Jin Oh
UC Berkeley
Blow-up and global dynamics for the self-dual Chern-Simons-Schrödinger model
The self-dual Chern-Simons-Schrödinger model is a gauged cubic NLS on the plane with self-duality, i.e., energy minimizers are given by a first-order Cauchy-Riemann-type equation, rather than a second-order 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 self-duality and nonlocality (which stems from the gauge structure). In accordance, this model possesses blow-up 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 blow-up and global dynamics of this model, with emphasis on a few surprising features of this model such as the impossibility of a bubble-tree'' blow-up and a nonlinear rotational instability of pseudoconformal blow-ups. 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 two-sphere. 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 continuously-in-time via a “no-return” lemma based on the virial identity. The proof combines a collision analysis of solutions near a multi-soliton configuration with concentration compactness techniques. As a byproduct of our analysis we also prove that there are no elastic collisions between pure multi-solitons.
February 8 Henrik Shahgholian
KTH
Global solutions to the obstacle problem and singular points
That ellipsoidal shells do not exert gravitational force inside the cavity of the shell was known to Newton, Laplace, and Ivory. In early 30’s P. Dive proved the inverse of this theorem. In this talk, I shall recall the (partially geometric) proof of this fact and then extend this result to unbounded domains. Since ellipsoids, and any limit of a sequence of ellipsoids, are the so-called coincidence sets for the obstacle problem, there is a close link between the ellipsoidal potential theory and global solutions to the obstacle problem. In this talk we present a complete classification (in terms of limit domains of ellipsoids) for global solutions to the obstacle problem in dimensions greater than five. The interesting ramification of this result is a new interpretation of the structure of the regular free boundary close to singular points. This is a joint work with S. Eberle, and G.S. Weiss. For further details and references see: https://www.scilag.net/problem/P-200218.1
February 15 Hung Tran
Periodic homogenization of Hamilton-Jacobi equations: optimal rate and finer properties
I will describe some recent progress in periodic homogenization of Hamilton-Jacobi equations. First, we show that the optimal rate of convergence is $O(\varepsilon)$ in the convex setting. We then give a minimalistic explanation that the class of centrally symmetric polygons with rational vertices and nonempty interior is admissible as effective fronts in two dimensions. Joint works with Wenjia Jing and Yifeng Yu.
February 22 Guido De Philippis
NYU

Postponed

March 8 Javier Gomez-Serrano
Brown University and University of Barcelona
Rigidity and flexibility of stationary solutions of the Euler equations
In this talk, I will discuss characterizations of stationary solutions of the 2D Euler equations in two different directions under different assumptions: rigidity (is every stationary solution radial?) and flexibility (do there exist non-radial stationary solutions?). The proofs will have a calculus of variations' flavor, a new observation that finite energy, stationary solutions with simply-connected vorticity have compactly supported velocity, and an application of the Nash-Moser iteration procedure. Based on joint work with Jaemin Park, Jia Shi and Yao Yao.

### Spring 2022

Date Speaker Title + Abstract
April 12 Roman Shvydkoy
UIC
Global hypocoercivity of Fokker-Planck-Alignment equations
In this talk we will discuss a new approach to the problem of emergence in hydrodynamic systems of collective behavior. The problem seeks to establish convergence to a flocking state in a system with self-organization governed by strictly local laws of communication. The typical results in this direction insist on propagation of flock connectivity which translates into a quantitative non-vacuum condition on macroscopic level. With the introduction of small noise one can relax such a condition considerably, and even allow for vacuum, in the context of the corresponding Fokker-Planck-Alignment equations. The flocking behavior becomes the problem of establishing hypocoercivity and relaxation of solutions to the global Maxwellian. We will describe a model which does precisely that in the non-perturbative settings.
May 3 James Kelliher
UC Riverside
3D Euler equations with inflow, outflow
The 3D incompressible Euler equations in a bounded domain are most often supplemented with impermeable boundary conditions, which constrain the fluid to neither enter nor leave the domain. In this talk, I will explain how we obtain well-posedness of solutions in which the full value of the velocity is specified on inflow and the normal component is specified on outflow. We do this for multiply connected domains, and establish compatibility conditions to obtain arbitrarily high Holder regularity. This is joint work with Gung-Min Gie and Anna Mazzucato.
May 17 Alexis Vasseur
UT Austin
Boundary vorticity estimate for the Navier-Stokes equation and control of layer separation in the inviscid limit
Consider the steady solution to the incompressible Euler equation $Ae_1$ in the periodic tunnel $\Omega=[0,1]\times \mathbb T^2$. Consider now the family of solutions $U_\nu$ to the associated Navier-Stokes equation with the no-slip condition on the flat boundaries, for small viscosities $\nu=1/ Re$, and initial values close in $L^2$ to $Ae_1$. Under a conditional assumption on the energy dissipation close to the boundary, Kato showed in 1984 that $U_\nu$ converges to $Ae_1$ when the viscosity converges to 0 and the initial value converges to $A e_1$. It is still unknown whether this inviscid limit is unconditionally valid. Actually, the convex integration method predicts the possibility of layer separation. It produces solutions to the Euler equation with initial values $Ae_1$, but with layer separation energy at time T up to $$\|U(T)-Ae_1\|^2_{L^2}\equiv A^3T.$$ In this work, we prove that at the double limit for the inviscid asymptotic $\bar{U}$, where both the Reynolds number $Re$ converges to infinity and the initial value $U_{\nu}$ converges to $Ae_1$ in $L^2$, the energy of layer separation cannot be more than $$\| \bar{U}(T)-Ae_1\|^2_{L^2}\lesssim A^3T.$$ Especially, it shows that, even if the limit is not unique, the shear flow pattern is observable up to time $1/A$. This provides a notion of stability despite the possible non-uniqueness of the limit predicted by the convex integration theory. The result relies on a new boundary vorticity estimate for the Navier-Stokes equation. This new estimate, inspired by previous work on higher regularity estimates for Navier-Stokes, provides a non-linear control scalable through the inviscid limit.
May 24 Benoit Perthame
Sorbonne University
Porous media based models of living tissues and free boundary problems
Tissue growth, as it occurs during solid tumors, can be described at a number of different scales from the cell to the organ. For a large number of cells, 'fluid mechanical' approaches have been advocated in mathematics, mechanics or biophysics. We will give an overview of the modeling aspects and focuss on the links between those mathematical models. Then, we will focus on the `compressible' description describing the cell population density based on systems of porous medium type equations with reaction terms. A more macroscopic 'incompressible' description is based on a free boundary problem close to the classical Hele-Shaw equation. In the stiff pressure limit, one can derive a weak formulation of the corresponding Hele-Shaw free boundary problem and one can make the connection with its geometric form. The mathematical tools related to these questions include multi-scale analysis, Aronson-Benilan estimate, compensated compactness, uniform $L^4$ estimate on the pressure gradient and emergence of instabilities.
May 31 Thomas Giletti
University of Lorraine
Travelling fronts in spatially periodic bistable and multistable equations
This talk will be devoted to the existence of pulsating travelling front solutions for spatially periodic heterogeneous reaction-diffusion equations in arbitrary dimension. In the bistable case, such a pulsating front indeed exists and it also describes the large time dynamics of solutions of the Cauchy problem. However, unlike in the homogeneous case the periodic problem is no longer invariant by rotation, so that the front speed may be different depending on its direction. This in turn raises some difficulties in the spreading shape of solutions of the evolution problem, which may exhibit strongly asymmetrical features. In the general multistable case, that is when there is a finite but arbitrary number of stable steady states, the notion of a single front is no longer sufficient and we instead observe the appearance of a so-called propagating terrace. This roughly refers to a finite family of stacked fronts connecting intermediate stable steady states and whose speeds are ordered. The presented results come from a series of work with W. Ding, A. Ducrot, H. Matano and L. Rossi.

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