Wasserstein stability of porous medium-type equations on manifolds with Ricci curvature bounded below

Abstract

Given a complete, connected Riemannian manifold Mn with Ricci curvature bounded from below, we discuss the stability of the solutions of a porous medium-type equation with respect to the 2-Wasserstein distance. We produce (sharp) stability estimates under negative curvature bounds, which to some extent generalize well-known results by Sturm [35] and Otto-Westdickenberg [32]. The strategy of the proof mainly relies on a quantitative L1–L∞ smoothing property of the equation considered, combined with the Hamiltonian approach developed by Ambrosio, Mondino and Savaré in a metric-measure setting [4].