Abstract:
Cross-diffusion systems arise naturally to describe the dynamics of multiple interacting species in the life and social sciences. Such systems can be formally derived from a simple exclusion process on a lattice or a stochastic differential equation system using for example the method of matched asymptotics. Often the underlying microscopic gradient flow structure is lost in the limit. Hence we introduce the concept of asymptotic gradient flows and discuss how it can be used to study the behavior close to equilibrium. Next we focus on segregation phenomena, which can often be observed in cross-diffusion systems. We discuss the analysis and the segregation process in the case of two models for pedestrian and population dynamics, and illustrate our results with numerical simulations.