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arXiv · 2111.13764

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

Romain Ducasse·Filippo Santambrogio·Havva Yoldaş
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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.

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Romain Ducasse, Filippo Santambrogio, Havva Yoldaş. 2022-10-05. A cross-diffusion system obtained via (convex) relaxation in the JKO scheme. https://doi.org/10.1007/s00526-022-02356-8

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