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Mini Workshop on PDEs

Introduction

This half-day mini-workshop aims at bring together researchers on partial differential equations. The topics include partial differential equations from fluid mechanics, geometry, and material sciences, etc. The mini-workshop is supported by National Science Foundational of China, Institute of Natural Sciences, and School of Mathematical Sciences in Shanghai Jiao Tong University.

Date

November 10, 2017

Venue

601 Pao Yue-Kong Library

Organizers

Chunjing Xie, Shanghai Jiao Tong University

Xiongfeng Yang, Shanghai Jiao Tong University

Schedule

Time Name Title
9:00-9:50 Shu Wang Quasi-neutral limit problems for drift-diffusion models in semiconductors
9:50-10:40 Jiguang Bao Monge-Ampere Equations in Exterior Domains
10:40-11:00 Tea Break  
11:00-11:50 Haigang Li Optimal estimates for elliptic equations and systems from composite material

Title and Abstract

Monge-Ampere Equations in Exterior Domains

Jiguang Bao, Beijing Normal University

In this talk, we survey the results on Jorgens-Calabi-Pogorelov theorems and the asymptotic behavior at infinity of the solution to elliptic Monge-Ampere equations. And then we ues level set and nonlinear perturbation method to obtain the asymptotic behavior at infinity of the solution to a kind of parabolic Monge-Ampere equations. Jorgens-Calabi-Pogorelov theorems for parabolic Monge-Ampere equation can be regarded as a special case of our results.

Optimal estimates for elliptic equations and systems from composite material

Haigang Li, Beijing Normal Unviersity

We study a class of second-order elliptic equations and systems of divergence form, with discontinuous coefficients and data, arising from the study of composite materials. For the original problem concerning the system of linear elasticity, we develop an iteration technique with respect to the energy integral to overcome the difficulty from the lack of maximal principles and obtain the optimal blow-up rates of the gradients when two inclusions are close to touch. Our results hold for convex inclusions with arbitrary shape and in all dimensions. For the scalar case, we first establish the explicit dependence of the gradient on the ellipticity coefficients and the distance between interfacial boundaries of inclusions, which unifies the known results in the literature and answers open problem (b) proposed by Li-Vogelius (2000). Secondly, more interesting higher-order derivative estimates are also obtained, answering open problem (c) of Li-Vogelius (2000). This is based on joint work with Professor Jiguang Bao(BNU), Hongjie Dong(Brown), and Yanyan Li(Rutgers).

Asymptotic theory for electromagnetic fluid dynamics system

Shu Wang, Beijing University of Technology

In this talk, we will introduce the asymptotic theory for electromagnetic fluid dynamics system in multiple fields coupling physical process. It includes the well-posedness, large time behavior, small parameter limit, and multiscale structural stability. We will survey the recent progress and establish global stability of multidimensional stationary solutions without smallness assumptions. We will give the rigorous justification between electromagnetic fluid and classical fluid.