Local Aronson-B\'enilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds

TL;DR

This paper develops local gradient and Laplacian estimates for porous medium and fast diffusion equations on manifolds with Ricci curvature bounds, and introduces new entropy formulae inspired by Perelman's work.

Contribution

It provides novel local Aronson-Bénilan and Li-Yau type estimates and entropy formulae for porous medium and fast diffusion equations on curved manifolds.

Findings

Derived local gradient and Laplacian estimates for solutions.

Established new entropy formulae inspired by Perelman.

Extended results to fast diffusion equations.

Abstract

In this work we derive local gradient and Laplacian estimates of the Aronson-B\'enilan and Li-Yau type for positive solutions of porous medium equations posed on Riemannian manifolds with a lower Ricci curvature bound. We also prove similar results for some fast diffusion equations. Inspired by Perelman's work we discover some new entropy formulae for these equations.