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. 

 

Degenerate diffusion in heterogeneous media. Long-time behaviour of solutions of the Cauchy problem

Speaker: 

Guillermo Reyes

Institution: 

UC Irvine

Time: 

Tuesday, November 20, 2012 - 3:00pm

Location: 

RH 440R

In this talk I will present some recent results concerning the
asymptotic self-similar patterns of degenerate diffusion in an infinite
porous medium with vanishing at infinity variable density.

The asymptotic pattern turns out to strongly depend on the decay rate of
the density. For "slowly" decaying densities, the picture is similar to
the homogeneous case (Barenblatt-type solutions), whereas for densities,
decaying fast enough, a completely different behavior, typical of problems
in bounded domains, arises.

For intermediate decay rates, both descriptions are correct, providing an
example of matched asymptotics.

Positive Equilibrium Solutions in Structured Population Dynamics

Speaker: 

Christoph Walker

Institution: 

Leibniz Universität Hannover

Time: 

Thursday, October 18, 2012 - 3:00pm

Location: 

RH 306

The talk focuses on positive equilibrium (i.e. time-independent) solutions to mathematical models for the dynamics of populations structured by age and spatial position. This leads to the study of quasilinear parabolic equations with nonlocal and possibly nonlinear initial conditions. We shall see in an abstract functional analytic framework how bifurcation techniques may be combined with optimal parabolic regularity theory to establish the existence of positive solutions. As an application of these results we give a description of the geometry of coexistence states in a two-parameter predator-prey model.

Contact Solutions for Fully Nonlinear PDE Systems and Applications to Calculus of Variations in $L^\infty$

Speaker: 

Nikolaos Katzourakis

Institution: 

Basque Center for Applied Mathematics, Spain

Time: 

Tuesday, November 6, 2012 - 3:00pm to 4:00pm

Host: 

Location: 

Rowland Hall 440R

We will introduce the rudiments of a new theory of non-smooth
solutions which applies to fully nonlinear PDE systems and extends
Viscosity Solutions of Crandall-Ishii-Lions to the general vector case.
Key ingredient is the discovery of a notion of Extremum for maps which
extends min-max uniquely and allows for ``nonlinear passage of
derivatives" to test maps. The notions supports uniqueness, existence
and stability results, preserving most features of the scalar viscosity
counterpart. We will also discuss applications in vector-valued Calculus
of Variations in $L^\infty$ and Hamilton-Jacobi PDE with vector
solution.

Selected topics in complex fluids

Speaker: 

Cheng Yu

Institution: 

Department of math, University of Pittsburgh

Time: 

Tuesday, October 16, 2012 - 3:00pm to 4:00pm

Host: 

Location: 

Rowland Hall 306

In this talk, we focus on the study of mathematical theory of
the complex fluids. In the first part, we discuss the global existence for
weak solutons to multidimensional compressible flow of nematic liquid
crystals and the incompressible limits. In the second part, we establish
global existence and uniqueness results for weak solutions to
multidimensional Navier-Stokes-Vlasov equations.

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