On nonlinear cross-diffusion systems: an optimal transport approach

TL;DR

This paper introduces a gradient flow framework for nonlinear cross-diffusion systems, analyzing their solutions, segregation behavior in one dimension, and the incompressible limit leading to Hele-Shaw flow.

Contribution

It develops a Wasserstein gradient flow approach for degenerate cross-diffusion models and establishes convergence and segregation results, including the Hele-Shaw limit in one dimension.

Findings

Discrete solutions derived from the gradient flow framework

Segregation and interface stability in one dimension

Incompressible limit leading to Hele-Shaw flow

Abstract

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.