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Flows and functional inequalities for fractional operators

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

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Type
Article accepté pour publication ou publié
Date
2017
Nom de la revue
Applicable Analysis
Volume
96
Numéro
9
Éditeur
Gordon and Breach
Pages
1547-1560
Identifiant publication
Métadonnées
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Auteur(s)
Dolbeault, Jean cc
Zhang, An
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