On nonlinear cross-diffusion systems: an optimal transport approach

摘要

We study a nonlinear, degenerate cross-diffusion model which involves twodensities with two different drift velocities. A general framework isintroduced based on its gradient flow structure in Wasserstein space to derivea notion of discrete-time solutions. Its continuum limit, due to the possiblemixing of the densities, only solves a weaker version of the original system.In one space dimension, where the densities are guaranteed to be segregated, astable interface appears between the two densities, and a stronger convergenceresult, in particular derivation of a standard weak solution to the system, isavailable. We also study the incompressible limit of the system, whichaddresses transport under a height constraint on the total density. In onespace dimension we show that the problem leads to a two-phase Hele-Shaw typeflow.