Nonlinear cross-diffusion systems: an optimal transport approach

Alpár Mészáros
Department of Mathematics, University of California, Los Angeles

In this talk we will present a degenerate cross-diffusion model which involves two densities with two different drift velocities. A general framework will be introduced based on its gradient flow structure in the Wasserstein space to derive a notion of discrete-time solutions. Its continuum limit, due to the possible mixing of the densities, only solves a weaker version of the original system. In one space dimension, where the densities are guaranteed to be segregated, a stable interface appears between the two densities, and a stronger convergence result, in particular derivation of a standard weak solution to the system, is available. We also study the incompressible limit of the system, which addresses transport under a height constraint on the total density. In one space dimension we show that the problem leads to a two-phase Hele-Shaw type flow. The talk is based on a joint work with Inwon Kim (UCLA).