22 July  to 23 July:  PDE workshop:

Chair Prof. Fanghua Lin and Jiaxing Hong

Second Week:.24 July -28  July

1,Professor Manuel Del Pino. Universidad de Chile

Title: Semi-linear elliptic PDEs, Sharp Interfaces and Related Geometric Variational Problems

Abstract: The lectures would be concentrated on  mainly on Gluing  methods for evolution problems. This is quite new and still on-going research program with numerous exciting developments. The lectures  will discuss constructions of Type II blow-ups for critical exponents semi-linear equations and harmonic map flows as well.

2, Professor Jun-Cheng Wei, UBC

 Title: Semi-linear elliptic PDEs, Sharp Interfaces and Related Geometric Variational Problems.

 Abstract:  The lectures would be focused on  Allen-Cahn equations, DeGiorgi conjecture,          gluing methods for higher dimensional concentrations, classification of         global, stable and   finite Morse index solutions.

3，Professor Tai-Chia  Lin， National Taiwan Univ.，

Title:  Ion Chanels, Kinetic Equations and Non-Standard Diffusions

Abstract:  Understanding ion transport is crucial in the study of many physical and biological problems, such as semiconductors, electro-kinetic fluids, and ion channels in cell membranes. One of the fundamental models for the ionic transport is the time dependent coupled diffusion-convection equations, the Poisson-Nernst-Planck (PNP) system. However, due to the lack of ionic finite size effects, conventional PNP systems cannot be used to study gating and selectivity of ion channels and new mathematical models need to be developed. In my lectures, I’ll introduce analytical techniques to derive and analyze PNP type systems for the study of ion transport through channels.

4， Professor  Chun  Liu，Pen. State Univ.

Title: Energetic Variational Approaches in Complex Fluids and Biology
Abstract: In this lecture series, we will introduce the general energetic variational approaches, which had been motivated by Rayleigh and Onsager. We will look at the applications in viscoelastic fluids, liquid crystals and polymeric fluids. If time permits, we will also
discuss the general diffusion, with applications in biology and physiology.

5.Professor  Hao Wu

Title: Energetic Variational Approaches in Complex Fluids and Electro-kinetic Systems

Abstract: We will discuss two issues that are closely related to the lectures Prof. Chun Liu and Tai-Chia Lin. 

(1) We show that the Poisson-Nernst-Planck (PNP) system can be justified as a macroscopic model for the transport of multi-species ions in

dilute solutions from the kinetic theory. Starting from a Vlasov�Poisson�Fokker�Planck (VPFP) system

in a bounded domain with reflection boundary conditions for charge distributions, we prove that the global renormalized solutions of the VPFP system converge to

the global weak solutions of the PNP system, as the small parameter related to the scaled thermal velocity and mean free path tends to zero. 

(2) We discuss the application of energetic variational approaches to the Cahn-Hilliard equation, which is a fundamental model for binary mixtures. We review results for the CH equation subject to different type of boundary conditions and its possible coupling with fluid equations. 

.

7.Ling Xiao

Translating Solitons in Euclidean Space. 

Abstract: Mean curvature flow may be regarded as a geometric version of the heat equation. However, in contrast to the classical heat equation, mean curvature flow is described by a quasilinear evolution system of partial differential equations, and in general the solution only exists on a finite time interval. Therefore, it's very typical that the flow develops singularities. 

Translating solitons arise as parabolic rescaling of type II singularities. In this talk, I shall outline a program on the classification of translating solitons. I shall also report on some recent progress we have made in the joint work with Joel Spruck.