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].
De Ponti, N., Muratori, M., Orrieri, C. (2022). Wasserstein stability of porous medium-type equations on manifolds with Ricci curvature bounded below. JOURNAL OF FUNCTIONAL ANALYSIS, 283(9) [10.1016/J.JFA.2022.109661].
Wasserstein stability of porous medium-type equations on manifolds with Ricci curvature bounded below
De Ponti, N;
2022
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].
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