Local Aronson–Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds

Abstract In this work we derive local gradient and Laplacian estimates of the Aronson–Benilan 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.