  In this work, we study the Wasserstein gradient flow of the Riesz energy
defined on the space of probability measures. The Riesz kernels define a
quadratic functional on the space of measure which is not in general
geodesically convex in the Wasserstein geometry, therefore one cannot conclude
to global convergence of the Wasserstein gradient flow using standard
arguments. Our main result is the exponential convergence of the flow to the
minimizer on a closed Riemannian manifold under the condition that the
logarithm of the source and target measures are H{\"o}lder continuous. To this
goal, we first prove that the Polyak-Lojasiewicz inequality is satisfied for
sufficiently regular solutions. The key regularity result is the global in-time
existence of H{\"o}lder solutions if the initial and target data are H{\"o}lder
continuous, proven either in Euclidean space or on a closed Riemannian
manifold. For general measures, we prove using flow interchange techniques that
there is no local minima other than the global one for the Coulomb kernel. In
fact, we prove that a Lagrangian critical point of the functional for the
Coulomb (or Energy distance) kernel is equal to the target everywhere except on
singular sets with empty interior. In addition, singular enough measures cannot
be critical points.
