Calculus of Variations and Geometric Measure Theory

N. De Ponti - M. Muratori - C. Orrieri

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

created by deponti on 09 Aug 2019
modified on 28 Jul 2022


Accepted Paper

Inserted: 9 aug 2019
Last Updated: 28 jul 2022

Journal: Journal of Functional Analysis
Year: 2019

ArXiv: 1908.03147 PDF


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.