  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
