本文選題：多孔介質(zhì)方程 + Ricci流　； 參考：《中國礦業(yè)大學(xué)》2015年碩士論文

【摘要】：在1986年,P Li和S.-T. Yau在黎曼流形上,得到了度量固定時,熱方程正解的梯度估計.并且在Ricci曲率非負(fù)的情況下,他們的估計是最優(yōu)的.后來,R.S.Hamilton得出了緊致黎曼流形上,熱方程的僅有空間導(dǎo)數(shù)的梯度估計.2009年,B ailesteanu-Cao-Pulemotov沿Ricci流對熱方程的正解,做出了Li-Yau型和Hamilton型梯度估計.受此啟發(fā),本文我們主要研究沿Ricci流的黎曼流形上多孔介質(zhì)方程正解的Li-Yau型和Hamilton型梯度估計.對于完備非緊的黎曼流形,我們得到局部的Li-Yau型梯度估計；對于緊致的黎曼流形,我們得出整體的Li-Yau型梯度估計.這些估計推廣了Bailesteanu-Cao-Pulemotov在[2]中對Ricci流下黎曼流形上熱方程的Li-Yau型局部及整體梯度估計和Hamilton型梯度估計.作為多孔介質(zhì)方程Li-Yau型梯度估計的應(yīng)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.
【學(xué)位授予單位】：中國礦業(yè)大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：O186.12

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 方守文;;延拓的Ricci流下一類熱方程正解的梯度估計[J];科技創(chuàng)新導(dǎo)報;2013年04期

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本文編號：2011064