Flows and functional inequalities for fractional operators

Abstract

This paper collects results concerning global rates and large time asymptotics of a fractional fast diffusion equation 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.

Keywords:

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

AMS Subject Classifications:

• Primary: 26A33
• 35R11
• Secondary: 35A23
• 35B33
• 35B40
• 35K65

Acknowledgements

The authors thank Maria J. Esteban for fruitful discussions and suggestions.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (A.Z., reference: ANR-10-LABX-0098, LabEx SMP) and by the projects STAB (J.D., A.Z.) and Kibord (J.D.) of the French National Research Agency (ANR). A.Z. thanks the ERC Advanced Grant BLOWDISOL (Blow-up, dispersion and solitons; PI: Frank Merle) [# 291214] for support.