# PDE Geometric Analysis seminar

The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

## Contents

- 1 Previous PDE/GA seminars
- 2 Tentative schedule for Fall 2015
- 3 Seminar Schedule Spring 2015
- 3.1 Abstracts
- 3.1.1 Jun Kitagawa (Toronto)
- 3.1.2 Jessica Lin (Madison)
- 3.1.3 Yaguang Wang (Shanghai Jiao Tong)
- 3.1.4 Benoit Pausader (Princeton)
- 3.1.5 Haozhao Li (University of Science and Technology of China)
- 3.1.6 Jennifer Beichman (UW-Madison)
- 3.1.7 Ben Fehrman (University of Chicago)
- 3.1.8 Vera Hur
- 3.1.9 Sung-Jin Oh
- 3.1.10 Yuan Lou

- 3.1 Abstracts

### Previous PDE/GA seminars

### Tentative schedule for Fall 2015

# Seminar Schedule Spring 2015

## Abstracts

### Jun Kitagawa (Toronto)

Regularity theory for generated Jacobian equations: from optimal transport to geometric optics

Equations of Monge-Ampere type arise in numerous contexts, and solutions often exhibit very subtle qualitative and quantitative properties; this is owing to the highly nonlinear nature of the equation, and its degeneracy (in the sense of ellipticity). Motivated by an example from geometric optics, I will talk about the class of Generated Jacobian Equations; recently introduced by Trudinger, this class also encompasses, for example, optimal transport, the Minkowski problem, and the classical Monge-Ampere equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field reflector problems (of interest from a physical and practical point of view). This talk is based on joint works with N. Guillen.

### Jessica Lin (Madison)

Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations

We establish error estimates for the stochastic homogenization of fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media. Based on the approach of Armstrong and Smart in the elliptic setting, we construct a quantity which captures the geometric behavior of solutions to parabolic equations. The error estimates are shown to be of algebraic order. This talk is based on joint work with Charles Smart.

### Yaguang Wang (Shanghai Jiao Tong)

Stability of Three-dimensional Prandtl Boundary Layers

In this talk, we shall study the stability of the Prandtl boundary layer equations in three space variables. First, we obtain a well-posedness result of the three-dimensional Prandtl equations under some constraint on its flow structure. It reveals that the classical Burgers equation plays an important role in determining this type of flow with special structure, that avoids the appearance of the complicated secondary flow in the three-dimensional Prandtl boundary layers. Second, we give an instability criterion for the Prandtl equations in three space variables. Both of linear and nonlinear stability are considered. This criterion shows that the monotonic shear flow is linearly stable for the three dimensional Prandtl equations if and only if the tangential velocity field direction is invariant with respect to the normal variable, which is an exact complement to the above well-posedness result for a special flow. This is a joint work with Chengjie Liu and Tong Yang.

### Benoit Pausader (Princeton)

Global smooth solutions for the Euler-Maxwell problem for electrons in 2 dimensions

It is well known that pure compressible fluids tend to develop shocks, even from small perturbation. We study how self consistent electromagnetic fields can stabilize these fluids. In a joint work with A. Ionescu and Y. Deng, we consider a compressible fluid of electrons in 2D, subject to its own electromagnetic field and to a field created by a uniform background of positively charged ions. We show that small smooth and irrotational perturbations of a uniform background at rest lead to solutions that remain globally smooth, in contrast with neutral fluids. This amounts to proving small data global existence for a system of quasilinear Klein-Gordon equations with different speeds.

### Haozhao Li (University of Science and Technology of China)

Regularity scales and convergence of the Calabi flow

We define regularity scales to study the behavior of the Calabi flow. Based on estimates of the regularity scales, we obtain convergence theorems of the Calabi flow on extremal K\"ahler surfaces, under the assumption of global existence of the Calabi flow solutions. Our results partially confirm Donaldson’s conjectural picture for the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra assumption that the scalar curvature is uniformly bounded.

### Jennifer Beichman (UW-Madison)

Nonstandard dispersive estimates and linearized water waves

In this talk, we focus on understanding the relationship between the decay of a solution to the linearized water wave problem and its initial data. We obtain decay bounds for a class of 1D dispersive equations that includes the linearized water wave. These decay bounds display a surprising growth factor, which we show is sharp. A further exploration leads to a result relating singularities of the initial data at the origin in Fourier frequency to the regularity of the solution.

### Ben Fehrman (University of Chicago)

On The Existence of an Invariant Measure for Isotropic Diffusions in Random Environments

I will discuss the existence of a unique mutually absolutely continuous invariant measure for isotropic diffusions in random environment, of dimension at least three, which are small perturbations of Brownian motion satisfying a finite range dependence. This framework was first considered in the continuous setting by Sznitman and Zeitouni and in the discrete setting by Bricmont and Kupiainen. The results of this talk should be seen as an extension of their work.

I will furthermore mention applications of this analysis to the stochastic homogenization of the related elliptic and parabolic equations with random oscillatory boundary data and, explain how the existence of an invariant measure can be used to prove a Liouville property for the environment. In the latter case, the methods were motivated by work in the discrete setting by Benjamini, Duminil-Copin, Kozma and Yadin.

### Vera Hur

Instabilities in nonlinear dispersive waves

I will speak on the wave breaking and the modulational instability of nonlinear wave trains in dispersive media. I will begin by a gradient blowup proof for the Boussinesq-Whitham equations for water waves. I will then describe a variational approach to determine instability to long wavelength perturbations for a general class of Hamiltonian systems, allowing for nonlocal dispersion. I will discuss KdV type equations with fractional dispersion in depth. Lastly, I will explain an asymptotics approach for Whitham's equation for water waves, qualitatively reproducing the Benjamin-Feir instability of Stokes waves.

### Sung-Jin Oh

Global well-posedness of the energy critical Maxwell-Klein-Gordon equation

The massless Maxwell-Klein-Gordon system describes the interaction between an electromagnetic field (Maxwell) and a charged massless scalar field (massless Klein-Gordon, or wave). In this talk, I will present a recent proof, joint with D. Tataru, of global well-posedness and scattering of this system for arbitrary finite energy data in the (4+1)-dimensional Minkowski space, in which the PDE is energy critical.

### Yuan Lou

Asymptotic behavior of the smallest eigenvalue of an elliptic operator and its applications to evolution of dispersal

We investigate the effects of diffusion and drift on the smallest eigenvalue of an elliptic operator with zero Neumann boundary condition. Various asymptotic behaviors of the smallest eigenvalue, as diffusion and drift rates approach zero or infinity, are derived. As an application, these qualitative results yield some insight into the evolution of dispersal in heterogeneous environments.