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

Abstract

Over the past fifteen 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 satisfying a more general criterion for uniqueness, motivated by the Osgood criterion for ordinary differential equations. We also prove the first quantitative estimates on convergence of the discrete gradient flow or "JKO scheme" outside of 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.

34 pages

References in corpus (3)

• Quasi-static evolution and congested crowd transport
• Congested aggregation via Newtonian interaction
• On the role of Riesz potentials in Poisson's equation and Sobolev embeddings

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• Exact periodic stripes for a local/nonlocal minimization problem with volume constraint
• Aggregation-Diffusion to Constrained Interaction: Minimizers & Gradient Flows in the Slow Diffusion Limit
• Newtonian repulsion and radial confinement: convergence towards steady state