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

We study gradient estimates for the positive solutions of the porous medium equations and the fast diffusion equationsassociated 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 mdimensional Bakry-Emery Ricci curvature.