Wasserstein-{\L}ojasiewicz inequalities and asymptotics of McKean-Vlasov equation

TL;DR

This paper establishes convergence to equilibrium for the McKean-Vlasov equation on the torus using Wasserstein-Łojasiewicz inequalities, even in nonconvex settings, broadening the understanding of gradient flow dynamics.

Contribution

It introduces a Wasserstein-Łojasiewicz gradient inequality approach for nonconvex McKean-Vlasov equations, enabling convergence analysis without convexity or log-Sobolev assumptions.

Findings

Proves convergence to equilibrium in nonconvex settings

Develops a Wasserstein-Łojasiewicz inequality framework

Applies to the Keller-Segel chemotaxis model

Abstract

We prove convergence to equilibrium for solutions to the McKean-Vlasov (granular media) equation on the flat torus in a genuinely nonconvex setting. Our approach is based on a Wasserstein-{\L}ojasiewicz gradient inequality for the associated free energy, established under mild analyticity assumptions on the confinement and interaction potentials. This yields convergence of the corresponding Wasserstein gradient flow without convexity assumption and without postulating log-Sobolev related functional inequalities. We expect this strategy to extend to more general nonconvex Wasserstein gradient flows. In the present work we develop it in the McKean-Vlasov setting, with the Keller-Segel chemotaxis model on the torus as a prominent application.