Calculus of Variations and Geometric Measure Theory

R. Ducasse - F. Santambrogio - H. Yoldas

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

created by santambro on 24 Feb 2022
modified by yoldas on 29 Sep 2023


Accepted Paper

Inserted: 24 feb 2022
Last Updated: 29 sep 2023

Journal: Calc. Var. PDEs
Year: 2022

ArXiv: 2111.13764 PDF


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.