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 Fall 2014

date	speaker	title	host(s)
September 15	Greg Kuperberg (UC-Davis)	Cartan-Hadamard and the Little Prince	Viaclovsky
September 22 (joint with Analysis Seminar)	Steve Hofmann (U. of Missouri)	Quantitative Rectifiability and Elliptic Equations	Seeger
Oct 6th,	Xiangwen Zhang (Columbia University)	Alexandrov's Uniqueness Theorem for Convex Surfaces	B.Wang
October 13	Xuwen Chen (Brown University)[1]	The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-body Evolution	C.Kim
October 20	Kyudong Choi (UW-Madison)	Finite time blow up for 1D models for the 3D Axisymmetric Euler Equations the 2D Boussinesq system	C.Kim
October 27	Chanwoo Kim (UW-Madison)	BV-Regularity of the Boltzmann Equation in Non-Convex Domains	Local
November 3	Myoungjean Bae (POSTECH)	Recent progress on study of Euler-Poisson system	M.Feldman
November 10	Philip Isett (MIT)	Hölder Continuous Euler Flows	C.Kim
November 17	Lei Wu	Geometric Correction for Diffusive Expansion in Neutron Transport Equation	C.Kim
November 24	Hongnian Huang (Univeristy of New Mexico)	TBA	B.Wang
December 1	Xuan Hien Nguyen (Iowa State University)	TBA	Angenent

Fall Abstracts

Greg Kuperberg

Cartan-Hadamard and the Little Prince.

Steve Hofmann

Quantitative Rectifiability and Elliptic Equations

A classical theorem of F. and M. Riesz states that for a simply connected domain in the complex plane with a rectifiable boundary, harmonic measure and arc length measure on the boundary are mutually absolutely continuous. On the other hand, an example of C. Bishop and P. Jones shows that the latter conclusion may fail, in the absence of some sort of connectivity hypothesis. In this talk, we discuss recent developments in an ongoing program to find scale-invariant, higher dimensional versions of the F. and M. Riesz Theorem, as well as converses. In particular, we discuss substitute results that continue to hold in the absence of any connectivity hypothesis.

Xiangwen Zhang

Alexandrov's Uniqueness Theorem for Convex Surfaces

A classical uniqueness theorem of Alexandrov says that: a closed strictly convex twice differentiable surface in R3 is uniquely determined to within a parallel translation when one gives a proper function of the principle curvatures. We will talk about a PDE proof for this thorem, by using the maximal principle and weak uniqueness continuation theorem of Bers-Nirenberg. Moreover, a stability result related to the uniqueness problem will be mentioned. This is a joint work with P. Guan and Z. Wang. If time permits, we will also briefly introduce the idea of our recent work on Alexandrov’s theorems for codimension two submanifolds in spacetimes.

Xuwen Chen

The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-body Evolution

We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is focusing and matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D N-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to infinity. We prove that the limiting structure of the density matrices counterbalances this diverging coefficient. We establish the convergence of the BBGKY sequence and hence the propagation of chaos for the focusing quantum many-body system. We derive rigorously the 1D focusing cubic NLS as the mean-field limit of this 3D focusing quantum many-body dynamic and obtain the exact 3D to 1D coupling constant.

Kyudong Choi

Finite time blow up for 1D models for the 3D Axisymmetric Euler Equations the 2D Boussinesq system

In connection with the recent proposal for possible singularity formation at the boundary for solutions of the 3d axi-symmetric incompressible Euler's equations / the 2D Boussinesq system (Luo and Hou, 2013), we study models for the dynamics at the boundary and show that they exhibit a finite-time blow-up from smooth data. This is joint work with T. Hou, A. Kiselev, G. Luo, V. Sverak, and Y. Yao.

Myoungjean Bae

Recent progress on study of Euler-Poisson system

In this talk, I will present recent progress on the following subjects: (1) Smooth transonic flow of Euler-Poisson system; (2) Transonic shock of Euler-Poisson system. This talk is based on collaboration with Ben Duan, Chujing Xie and Jingjing Xiao

Philip Isett

"Hölder Continuous Euler Flows"

Motivated by the theory of hydrodynamic turbulence, L. Onsager conjectured in 1949 that solutions to the incompressible Euler equations with Holder regularity less than 1/3 may fail to conserve energy. C. De Lellis and L. Székelyhidi, Jr. have pioneered an approach to constructing such irregular flows based on an iteration scheme known as convex integration. This approach involves correcting “approximate solutions" by adding rapid oscillations which are designed to reduce the error term in solving the equation. In this talk, I will discuss an improved convex integration framework, which yields solutions with Holder regularity 1/5-, as well as other related results.

Lei Wu

Geometric Correction for Diffusive Expansion in Neutron Transport Equation

We revisit the diffusive limit of a steady neutron transport equation in a 2-D unit disk with one-speed velocity. The traditional method is Hilbert expansions and boundary layer analysis. We will carefully study the classical theory of the construction of boundary layers, and discuss the necessity and specific method to add the geometric correction.

Xuan Hien Nguyen

"TBA"