The evolution variational inequality for weighted Wasserstein metrics in non-convex bounded domains

TL;DR

This paper extends the evolution variational inequality for weighted Wasserstein distances to non-convex domains, enabling analysis of complex PDEs like Keller--Segel and Cahn--Hilliard systems without convexity assumptions.

Contribution

It removes the convexity requirement by controlling boundary integrals, broadening the applicability of weighted Wasserstein metrics in PDE analysis.

Findings

Established evolution variational inequality in non-convex domains.

Applied inequality to obtain weak solutions of Keller--Segel and Cahn--Hilliard equations.

Developed boundary control techniques using Sobolev trace and Kato's inequality.

Abstract

In this paper, we establish the evolution variational inequality for the weighted Wasserstein distance, without assuming convexity of domains. Thanks to this evolution variational inequality, we can carry out some arguments with weighted Wasserstein metrics in not only convex but also non-convex domains. Therefore finally, we apply the evolution variational inequality to the minimizing movement in weighted Wasserstein metrics to obtain weak solutions of Keller--Segel systems and Cahn--Hilliard type equations in non-convex domains. The key point to remove the convexity assumption is a control of the boundary integral. To deal with the boundary integral, we use estimates for functions on the boundary, the Sobolev trace embedding and the variant of Kato's inequality. Then, the boundary integral can be absorbed by good known terms.