Flows and functional inequalities for fractional operators

Dolbeault, Jean; Zhang, An

Dolbeault, Jean; Zhang, An (2017), Flows and functional inequalities for fractional operators, Applicable Analysis, 96, 9, p. 1547-1560. 10.1080/00036811.2017.1286647

Voir/Ouvrir

DZ2016b.pdf (314.3Kb)

Type

Article accepté pour publication ou publié

Date

2017

Nom de la revue

Applicable Analysis

Volume

Numéro

Éditeur

Gordon and Breach

Pages

1547-1560

Résumé (EN)

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

self-similar variables; 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