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 on 02 Oct 2022


Accepted Paper

Inserted: 24 feb 2022
Last Updated: 2 oct 2022

Journal: Calc. Var. PDEs
Year: 2022


In this paper, we start from a very natural system of cross-diffusion equations, which can be seen as the gradient flow for the Wasserstein distance of a certain functional. Unfortunately, the cross-diffusion system is not well-posed, this is a consequence of the fact that the underlying functional is not lower semi-continuous. We then consider the relaxation of the functional, and we 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.