  In this paper we study gradient estimates for the positive solutions of the
porous medium equation: $$u_t=\Delta u^m$$ where $m>1$, which is a nonlinear
version of the heat equation. We derive local gradient estimates of the Li-Yau
type for positive solutions of porous medium equations on Riemannian manifolds
with Ricci curvature bounded from below. As applications, several parabolic
Harnack inequalities are obtained. In particular, our results improve the ones
of Lu, Ni, V\'{a}zquez and Villani in [10]. Moreover, our results recover the
ones of Davies in [4], Hamilton in [5] and Li and Xu in [7].
