Nonconvex gradient flow in the Wasserstein metric and applications to constrained nonlocal interactions

Abstract

Over the past 15 years, the theory of Wasserstein gradient flows of convex (or, more generally, semiconvex) energies has led to advances in several areas of partial differential equations and analysis. In this work, we extend the well-posedness theory for Wasserstein gradient flows to energies that are merely ω-convex. This new criterion for uniqueness is a generalization of the classical notion of semiconvexity along geodesics, motivated by the Osgood criterion for uniqueness of ordinary differential equations. Under this assumption, we go on to prove the first quantitative estimates on convergence of the discrete gradient flow or JKO scheme outside the semiconvex case. We conclude by applying these results to study the well-posedness of constrained nonlocal interaction energies, which have arisen in recent work on biological chemotaxis and congested aggregation.