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

Guangyue Huang and Haizhong Li

Vol. 268 (2014), No. 1, 47–78

DOI: 10.2140/pjm.2014.268.47

Abstract

We study gradient estimates for the positive solutions of the porous medium equations and the fast diffusion equations

$$u_t={\Delta }_{\varphi }\left(u^p\right)$$

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 some gradient estimates which generalize some previous results of Lu et. al. and Huang et. al. As applications, several parabolic Harnack inequalities are obtained. Moreover, inspired by X.-D. Li’s work, we also extend the entropy formulae introduced by Lu et. al. to the porous medium equations and the fast diffusion equations associated with the Witten Laplacian. We prove some monotonicity theorems for such entropy on compact Riemannian manifolds with nonnegative $m$-dimensional Bakry–Emery Ricci curvature.

Keywords

porous medium equation, fast diffusion equation, entropy formulae, Witten Laplacian

Mathematical Subject Classification 2010

Primary: 35B45

Secondary: 35K55

Milestones

Received: 10 December 2012

Revised: 13 May 2013

Accepted: 23 July 2013

Published: 21 May 2014