Entropy methods and non-linear diffusion equations: some recent results

Jean Dolbeault
Universite Paris Dauphine
email: dolbeaul@ceremade.dauphine.fr

Abstract: The goal of this talk is to present a few recent results on entropy methods for non-linear diffusion equations. After briefly recalling how Gagliardo-Nirenberg inequalities enter in the study of intermediate asymptotics for the fast diffusion equation, new results based on Hardy-Poincare inequalities will be introduced. This connection with a linear setting raises the question of other Lyapunov functionals. Partial answers will be given for one-dimensional periodic solutions of equations involving second and fourth order operators. As an additional contribution, results based on capacity-measure criteria will be shown, with applications to weighted porous media equations.