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

## TL;DR

This paper establishes sharp stability estimates for solutions of porous medium-type equations on manifolds with Ricci curvature bounded below, extending previous results using a combination of smoothing properties and Hamiltonian methods.

## Contribution

It provides new stability estimates for porous medium equations on curved manifolds, generalizing prior work by Sturm and Otto-Westdickenberg.

## Key findings

- Sharp stability estimates under negative curvature bounds
- Extension of known results to more general manifolds
- Use of Hamiltonian approach in a metric-measure setting

## 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.