Gradient estimates and entropy formulae of porous medium and fast diffusion equations for the Witten Laplacian

TL;DR

This paper derives gradient estimates and entropy formulae for solutions to porous medium and fast diffusion equations involving the Witten Laplacian on Riemannian manifolds, extending previous results under curvature bounds.

Contribution

It generalizes gradient estimates and entropy formulae for these equations on manifolds with Bakry-Emery Ricci curvature bounds, inspired by prior work.

Findings

Gradient estimates for positive solutions under curvature bounds

Monotonicity of entropy formulae on compact manifolds

Extension of previous results to Witten Laplacian setting

Abstract

We consider gradient estimates to positive solutions of porous medium equations and fast diffusion equations: $$u_t=\Delta_\phi(u^p)$$ associated with the Witten Laplacian on Riemannian manifolds. Under the assumption that the $m$-dimensional Bakry-Emery Ricci curvature is bounded from below, we obtain gradient estimates which generalize the results in [20] and [13]. Moreover, inspired by X. -D. Li's work in [19] we also study the entropy formulae introduced in [20] for porous medium equations and fast diffusion equations associated with the Witten Laplacian. We prove monotonicity theorems for such entropy formulae on compact Riemannian manifolds with non-negative $m$-dimensional Bakry-Emery Ricci curvature