GRADIENT ESTIMATES AND ENTROPY FORMULAE OF POROUS MEDIUM AND FAST DIFFUSION EQUATIONS FOR THE WITTEN LAPLACIAN
GRADIENT ESTIMATES AND ENTROPY FORMULAE OF POROUS MEDIUM AND FAST DIFFUSION EQUATIONS FOR THE WITTEN LAPLACIAN
复制标题
多孔介质的梯度估计和熵公式以及维滕拉普拉斯的快速扩散方程
DOI：
10.2140/pjm.2014.268.47
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发表时间：
2014-03-01
影响因子：
0.6
通讯作者：
Li, Haizhong
中科院分区：
文献类型：
--
作者：
Huang, Guangyue;Li, Haizhong
我们研究与黎曼流形上的维滕拉普拉斯算子相关的多孔介质方程和快速扩散方程su(t) = Delta(phi)(u(p))的正解的梯度估计。在 m 维 Bakry-Emery Ricci 曲率有界于下的假设下，我们获得了一些梯度估计，这些估计概括了 Lu 等人之前的一些结果。等人。和黄等人。等人。作为应用，得到了几个抛物线哈纳克不等式。此外，受到 X.-D 的启发。在 Li 的工作中，我们还扩展了 Lu 等人引入的熵公式。等人。多孔介质方程和与维滕拉普拉斯相关的快速扩散方程。我们在具有非负 m 维 Bakry-Emery Ricci 曲率的紧致黎曼流形上证明了此类熵的一些单调性定理。
We study gradient estimates for the positive solutions of the porous medium equations and the fast diffusion equationsu(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 some gradient estimates which generalize some previous results of Lu et. al. and Huang et. al. As applications, several parabolic Harnack inequalities are obtained. Moreover, inspired by X.-D. Li's work, we also extend the entropy formulae introduced by Lu et. al. to the porous medium equations and the fast diffusion equations associated with the Witten Laplacian. We prove some monotonicity theorems for such entropy on compact Riemannian manifolds with nonnegative m-dimensional Bakry-Emery Ricci curvature.