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In this work we derive local gradient and Laplacian estimates of theAronson-B\'enilan and Li-Yau type for positive solutions of porous mediumequations posed on Riemannian manifolds with a lower Ricci curvature bound. Wealso prove similar results for some fast diffusion equations. Inspired byPerelman's work we discover some new entropy formulae for these equations.

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