APDE: Seminars

Seminars in Applied and Computational Analysis

On asymptotic gradient flow structures of PDE models with excluded volume effects

We discuss the analysis of a cross-diffusion PDE system for mixtures of hard spheres, which was derived by Bruna and Chapman (J Chem Phys 137, 2012) from a stochastic system of interacting Brownian particles using the methods of matched asymptotics. While the system has a gradient flow structure in the symmetric case of all particles having the same size and diffusivity, this is not valid in general. For the general case, we introduce the concept of an asymptotic gradient flow structure and show how it can be used to study the behavior close to equilibrium. To gain further insights into the dynamics of asymptotic gradient flows, we study the system in the special case of two specific species – namely diffusing (Brownian) and immobile (obstacle) particles. In this case the cross-diffusion system reduces to a single nonlinear nonlinear Fokker—Planck equation, which again has no full gradient flow structure. However it can be interpreted as an asymptotic gradient flow for different entropy and mobility pairs. We discuss several possible such pairs and present global in time existence results as well as study the long time behavior of the corresponding full gradient flow equation. Furthermore we illustrate the dynamics of the different equations with numerical simulations.

This is joint work with M. Bruna (Oxford), M. Burger (Münster) and H. Ranetbauer (Vienna)

Dispersive Quantization of Linear and Nonlinear Waves

The evolution, through spatially periodic linear dispersion, of rough initial data leads to surprising quantized structures at rational times, and fractal, non-differentiable profiles at irrational times. The Talbot effect, named after an optical experiment by one of the founders of photography, was first observed in optics and quantum mechanics, and leads to intriguing connections with exponential sums arising in number theory. Ramifications of these phenomena and recent progress on the analysis, numerics, and extensions to nonlinear wave models will be discussed.

Nonsmooth optimization based on piecewise linearization

Abstract not available

Graduate Seminars (WE 15:00, MR5)