Swarm dynamics and equilibria for a nonlocal aggregation model

Speaker: 

Razvan Fetecau

Institution: 

Simon Fraser University

Time: 

Tuesday, March 5, 2013 - 3:00pm

Location: 

RH 306

 

We consider the aggregation equation ρt − ∇ · (ρ∇K ∗ ρ) = 0 in Rn, where the interaction potential K models short-range repulsion and long-range attraction. We study a family of interaction potentials with repulsion given by a Newtonian potential and attraction in the form of a power law. We show global well-posedness of solutions and investigate analytically and numerically the equilibria and their global stability. The equilibria have biologically relevant features, such as finite densities and compact support with sharp boundaries. This is joint work with Yanghong Huang and Theodore Kolokolnikov. 

 

Homogenization of Green and Neumann Functions

Speaker: 

Zhongwei Shen

Institution: 

University of Kentucky

Time: 

Tuesday, March 12, 2013 - 3:00pm to 4:00pm

Host: 

Location: 

306RH

 

In the talk I will describe my recent work, joint with Carlos Kenig and
Fanghua Lin, on homogenization of the Green and Neumann functions for a family of second order elliptic systems with highly oscillatory periodic coefficients. We study the asymptotic behavior of the first derivatives of the Green and Neumann functions, using Dirichlet and Neumann correctors. As a result, we obtain asymptotic expansions of Poisson kernels and the Dirichlet-to-Neumann maps as well as optimal convergence rates in L^p and W^{1,p} for solutions with Dirichlet or Neumann boundary conditions.

Nonlinear Wave Equations With Damping And Supercritical Sources

Speaker: 

Yanqiu Guo

Institution: 

Weizman Institute

Time: 

Tuesday, January 8, 2013 - 2:00pm

Location: 

RH 340P

In this talk I will discuss the local and global well-posedness of coupled non- linear wave equations with damping and supercritical sources. Our interests lie in the interaction between source and damping terms and their influence on the behavior of solutions. I will introduce the method of using the monotone operator theory to obtain the local existence of weak solutions to our system. Also we extend a result by Brezis on convex integrals on Sobolev spaces, which allows us to overcome a major technical difficulty in the proof of the existence of solutions. 

 

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