Flows and functional inequalities for fractional operators

TL;DR

This paper studies the long-term behavior and inequalities related to fractional fast diffusion equations, highlighting differences from classical diffusion and exploring stability and entropy methods.

Contribution

It provides new insights into the asymptotics, stability, and entropy techniques for fractional diffusion, contrasting with classical cases.

Findings

Self-similar solutions are generally not optimal for fractional inequalities.

Entropy methods reveal stability properties and raise open problems.

Connections between fractional diffusion and Gagliardo-Nirenberg-Sobolev inequalities are established.

Abstract

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 carr{\'e} du champ and R{\'e}nyi entropy powers methods are investigated and raise a number of open problems.