The Gradient Estimates For Porous Medium Equations Under The Ricci Flow

In 1986, P. Li and S.-T. Yau got gradient estimate for positive solutions of the heat equations on the Riemann manifolds when the metric is fixed. In addition, this estimate that they got is optimal if Ricci curvature of the Riemann manifold is nonnegative. Later, R.S.Hamilton established gradient estimates involving only space derivative of heat equations on the compact Riemann manifold. Bailesteanu-Cao-Pulemotov proved the Li-Yau type and Hamilton type gradient estimates for positive solutions of the heat equations when the metric is involved under the Ricci flow in 2009.Inspired by these results, in this thesis we are mainly concerned with the Li-Yau type and Hamilton type gradient estimates for positive solutions of porous medium equations on Riemann manifolds with the metrics involving the Ricci flow. If the Riemann manifolds are complete and noncompact, we get a local Li-Yau type estimate. Moreover, if the Riemann manifolds are compact, we get a global Li-Yau type estimate. These estimates generalize the local and the global gradient estimates of the Li-Yau type and Hamilton type by Bailesteanu-Cao-Pulemotov [2] for positive solutions of the heat equations on the Riemann manifolds under the Ricci flow.As application of Li-Yau type gradient estimates for PME, we obtain Harnack inequalities in this thesis.