A cross-diffusion system obtained via (convex) relaxation in the JKO scheme

TL;DR

This paper develops a relaxed gradient flow approach for a cross-diffusion system derived from a non-lower semi-continuous functional, establishing existence of solutions despite the system's initial ill-posedness.

Contribution

It introduces a relaxation of the functional to obtain a well-posed gradient flow for a cross-diffusion system, addressing issues of non-uniqueness and ill-posedness.

Findings

Existence of solutions for the relaxed gradient flow.

The relaxed system retains a cross-diffusion structure.

The mixture of regimes complicates the analysis but is manageable.

Abstract

In this paper, we start from a very natural system of cross-diffusion equations, which can be seen as a gradient flow for the Wasserstein distance of a certain functional. Unfortunately, the cross-diffusion system is not well-posed, as a consequence of the fact that the underlying functional is not lower semi-continuous. We then consider the relaxation of the functional, and prove existence of a solution in a suitable sense for the gradient flow of (the relaxed functional). This gradient flow has also a cross-diffusion structure, but the mixture between two different regimes, that are determined by the relaxation, makes this study non-trivial.