On some non linear evolution systems which are perturbations of Wasserstein gradient flows

TL;DR

This paper establishes existence and uniqueness for certain nonlinear parabolic systems with nonlocal interactions, extending Wasserstein gradient flow results from periodic settings to the entire space and bounded domains.

Contribution

It extends the theory of Wasserstein gradient flows to more general non-periodic and bounded domain settings for nonlinear systems.

Findings

Existence of solutions via semi-implicit JKO scheme

Uniqueness from displacement convexity

Extension from periodic to bounded and unbounded domains

Abstract

This paper presents existence and uniqueness results for a class of parabolic systems with non linear diffusion and nonlocal interaction. These systems can be viewed as regular perturbations of Wasserstein gradient flows. Here we extend results known in the periodic case to the whole space and on a smooth bounded domain. Existence is obtained using a semi-implicit Jordan-Kinderlehrer-Otto scheme and uniqueness follows from a displacement convexity argument.