17 Febbraio, 2012 14:00 oclock

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Free energies, nonlinear flows and functional inequalities

Jean Dolbeault, Universite Paris Dauphine

Aula Seminari III piano

This lecture will primarily be devoted to a review of results based on entropy methods in nonlinear diffusion equations. The basic example is the fast diffusion equation in the euclidean space and the study of the asymptotic behaviour of the solutions in self-similar variables. Recent results (in collaboration with G. Toscani) provide interesting refinements for the study of the asymptotic behaviour of the solutions, based on best matching asymptotic profiles rather than on self-similar rescalings. As a consequence, improved Sobolev inequalities have been obtained, thus giving an answer to an old open question raised by H. Brezis and E. Lieb. Nonlocal improvements of standard functional inequalities will also be introduced, based on duality and nonlinear flows approaches. They are connected with mean field models like the Keller-Segel system.