  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.
