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

Details 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

2012-03-25

arxiv.org/abs/1203.5482v1

Mathematics

Differential Geometry

25 pages

Scientific paper

We consider gradient estimates to positive solutions of porous medium equations and fast diffusion equations: $$u_t=\Delta_\phi(u^p)$$ 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 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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