Calculus of Variations and Geometric Measure Theory
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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 31 Aug 2020



Inserted: 9 aug 2019
Last Updated: 31 aug 2020

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.

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