Accepted Paper

Inserted: 9 aug 2019
Last Updated: 28 jul 2022

Journal: Journal of Functional Analysis
Year: 2019

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 Savar\'e in a metric-measure setting.

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