Noemie David (Lyon)

New Lipschitz estimates and long-time asymptotic behavior for porous medium and fast diffusion equations. .

Among the class of nonlinear diffusion equations, the porous medium equation has certainly attracted major interest and its regularity theory is nowadays well established and understood. In this talk, I will present some recent results based on a joint work with F. Santambrogio. 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.