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.

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 16
February 17 (joint with Analysis Seminar: 4PM, B139)	Chanwoo Kim (Madison)	Hydrodynamic limit from the Boltzmann to the Navier-Stokes-Fourier	Seeger
February 23	Jennifer Beichman (Madison)	TBA	Kim
March 2	Benoit Pausader (Princeton)	TBA	Kim
March 9	Haozhao Li (University of Science and Technology of China)	Regularity scales and convergence of the Calabi flow	Wang
March 16
March 23	Ben Fehrman (University of Chicago)	TBA	Lin
March 30	Spring recess Mar 28-Apr 5 (S-N)
April 6
April 13
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.

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.