沿Ricci流的多孔介质方程的梯度估计

发布时间：2018-06-12 20:56

本文选题：多孔介质方程 + Ricci流　；参考：《中国矿业大学》2015年硕士论文

【摘要】：在1986年,P Li和S.-T. Yau在黎曼流形上,得到了度量固定时,热方程正解的梯度估计.并且在Ricci曲率非负的情况下,他们的估计是最优的.后来,R.S.Hamilton得出了紧致黎曼流形上,热方程的仅有空间导数的梯度估计.2009年,B ailesteanu-Cao-Pulemotov沿Ricci流对热方程的正解,做出了Li-Yau型和Hamilton型梯度估计.受此启发,本文我们主要研究沿Ricci流的黎曼流形上多孔介质方程正解的Li-Yau型和Hamilton型梯度估计.对于完备非紧的黎曼流形,我们得到局部的Li-Yau型梯度估计；对于紧致的黎曼流形,我们得出整体的Li-Yau型梯度估计.这些估计推广了Bailesteanu-Cao-Pulemotov在[2]中对Ricci流下黎曼流形上热方程的Li-Yau型局部及整体梯度估计和Hamilton型梯度估计.作为多孔介质方程Li-Yau型梯度估计的应用,本文得到了Harnack不等式.
[Abstract]:In 1986, on Riemannian manifolds, the gradient estimates of the positive solutions of the heat equation were obtained for P Li and S. T.Yau on Riemannian manifolds. In the case of non-negative Ricci curvature, their estimates are optimal. Then R. S. Hamilton obtained the gradient estimate of the thermal equation with only spatial derivative on the compact Riemannian manifold. In 2009, the positive solution of the heat equation was obtained by Bailesteanu-Cao-Pulemotov along the Ricci flow, and the Li-Yau type and Hamilton type gradient estimates were obtained. Inspired by this, we mainly study the Li-Yau type and Hamiltonian gradient estimates of the positive solutions of porous media equations on Riemannian manifolds along Ricci flows. For complete non-compact Riemannian manifolds, we obtain local Li-Yau type gradient estimators, and for compact Riemannian manifolds, we obtain global Li-Yau type gradient estimates. These estimates generalize the Li-Yau type local and global gradient estimates and Hamiltonian gradient estimates for the heat equations on Riemannian manifolds under Ricci flow in Bailesteanu-Cao-Pulemotov [2]. As an application of Li-Yau type gradient estimation for porous medium equations, Harnack inequality is obtained in this paper.
【学位授予单位】：中国矿业大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O186.12

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