Local Aronson-B\'enilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds

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Lu, Peng
Ni, Lei
Vázquez, Juan-Luis
Villani, Cédric
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Abstract
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In this work we derive local gradient and Laplacian estimates of the Aronson-B\'enilan and Li-Yau type for positive solutions of porous medium equations posed on Riemannian manifolds with a lower Ricci curvature bound. We also prove similar results for some fast diffusion equations. Inspired by Perelman's work we discover some new entropy formulae for these equations.
Comment: 22 pages, no figure
Keywords
Mathematics - Differential Geometry, Mathematics - Analysis of PDEs, 53C44, 58J35
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