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Wasserstein-Łojasiewicz inequalities and asymptotics of McKean-Vlasov equation

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 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.