  We study a nonlinear, degenerate cross-diffusion model which involves two
densities with two different drift velocities. A general framework is
introduced based on its gradient flow structure in 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.
