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

Abstract

Given a complete, connected Riemannian manifold ${\mathbb{M}}^n$ 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 and Otto-Westdickenberg. The strategy of the proof mainly relies on a quantitative $L^1-L^{\infty }$ smoothing property of the equation considered, combined with the Hamiltonian approach developed by Ambrosio, Mondino and Savaré in a metric-measure setting.

41 pages. We have corrected the definition of the nondegenerate nonlinearity. Fixed some typos. Final version, to appear in "Journal of Functional Analysis"

References in corpus (1)

• Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds