Local Aronson-B\'enilan estimates and entropy formulae for porous medium   and fast diffusion equations on manifolds
  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. 
Peng Lu, Lei Ni, Juan-Luis V\'azquez and C\'edric Villani