Wasserstein-Łojasiewicz inequalities and asymptotics of McKean-Vlasov equation

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-Łojasiewicz 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 any convexity assumptions 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 key application.