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
Previous PDE/GA seminars
Tentative schedule for Fall 2015
Seminar Schedule Spring 2015
date | speaker | title | host(s) |
---|---|---|---|
January 21 (Departmental Colloquium: 4PM, B239) | Jun Kitagawa (Toronto) | Regularity theory for generated Jacobian equations: from optimal transport to geometric optics | Feldman |
February 9 | Jessica Lin (Madison) | Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations | Kim |
February 17 (Tuesday) (joint with Analysis Seminar: 4PM, B139) | Chanwoo Kim (Madison) | Hydrodynamic limit from the Boltzmann to the Navier-Stokes-Fourier | Seeger |
February 23 (special time*, 3PM, B119) | Yaguang Wang (Shanghai Jiao Tong) | Stability of Three-dimensional Prandtl Boundary Layers | Jin |
March 2 *CANCELED* | Benoit Pausader (Princeton) | Global smooth solutions for the Euler-Maxwell problem for electrons in 2 dimensions *CANCELED* | Kim |
March 9 | Haozhao Li (University of Science and Technology of China) | Regularity scales and convergence of the Calabi flow | Wang |
March 16 | Jennifer Beichman (Madison) | Kim | |
March 23 | Ben Fehrman (University of Chicago) | TBA | Lin |
March 30 | Spring recess Mar 28-Apr 5 (S-N) | ||
April 6 | Vera Hur (UIUC) | Yao | |
April 13 | Sung-Jin Oh (Berkeley) | ||
April 20 | Yuan Lou (Ohio State) | TBA | Zlatos |
April 27 | |||
May 4 |
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.