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arXiv · 1203.5482

Gradient estimates and entropy formulae of porous medium and fast diffusion equations for the Witten Laplacian

Guangyue Huang·Haizhong Li
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Abstract

We consider gradient estimates to positive solutions of porous medium equations and fast diffusion equations: $$u_t=Δ_ϕ(u^p)$$ 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 gradient estimates which generalize the results in [20] and [13]. Moreover, inspired by X. -D. Li's work in [19] we also study the entropy formulae introduced in [20] for porous medium equations and fast diffusion equations associated with the Witten Laplacian. We prove monotonicity theorems for such entropy formulae on compact Riemannian manifolds with non-negative $m$-dimensional Bakry-Emery Ricci curvature

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Guangyue Huang, Haizhong Li. 2012-03-25. Gradient estimates and entropy formulae of porous medium and fast diffusion equations for the Witten Laplacian. https://arxiv.org/abs/1203.5482

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