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.