Flows and functional inequalities for fractional operators

Jean Dolbeault (1) , An Zhang (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This paper collects results concerning global rates and large time asymptotics of a fractional fast diffusion on the Euclidean space, which is deeply related with a family of fractional Gagliardo-Nirenberg-Sobolev inequalities. Generically, self-similar solutions are not optimal for the Gagliardo-Nirenberg-Sobolev inequalities, in strong contrast with usual standard fast diffusion equations based on non-fractional operators. Various aspects of the stability of the self-similar solutions and of the entropy methods like carré du champ and Rényi entropy powers methods are investigated and raise a number of open problems.

Mots clés: Rényi entropy powers; carré du champ; fractional Gagliardo-Nirenberg-Sobolev inequality; fractional Sobolev inequality; fractional fast diffusion equation; self-similar solutions; asymptotic behavior; intermediate asymptotics; rate of convergence; entropy methods; entropy – entropy production inequality; self-similar variables

Domaines: Equations aux dérivées partielles [math.AP]

Jean Dolbeault, An Zhang. Flows and functional inequalities for fractional operators. Applicable Analysis, 2017, 96 (9), pp.1547-1560. ⟨hal-01404580⟩