$\mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds

TL;DR

This paper develops Perelman type $ abla$-entropy formulas and differential Harnack estimates for porous medium equations on Riemannian manifolds, leading to new inequalities and estimates for solutions.

Contribution

It introduces novel $ abla$-entropy formulas and differential Harnack estimates specifically for porous medium equations on curved manifolds.

Findings

Established Perelman type $ abla$-entropy formulas

Derived global differential Harnack estimates

Obtained Harnack inequalities and Laplacian estimates

Abstract

In this paper, we prove Perelman type $\mathcal{W}$-entropy formulae and global differential Harnack estimates for positive solutions to porous medium equation on the closed Riemannian manifolds with Ricci curvature bounded below. As applications, we derive Harnack inequalities and Laplacian estimates.