  We obtain new estimates for the solution of both the porous medium and the
fast diffusion equations by studying the evolution of suitable Lipschitz norms.
Our results include instantaneous regularization for all positive times,
long-time decay rates of the norms which are sharp and independent of the
initial support, and new convergence results to the Barenblatt profile.
Moreover, we address nonlinear diffusion equations including quadratic or
bounded potentials as well. In the slow diffusion case, our strategy requires
exponents close enough to 1, while in the fast diffusion case, our results
cover any exponent for which the problem is well-posed and mass-preserving in
the whole space.
