A functional framework for the Keller-Segel system: logarithmic Hardy-Littlewood-Sobolev and related spectral gap inequalities

TL;DR

This paper develops a functional framework using logarithmic Hardy-Littlewood-Sobolev inequalities to analyze the Keller-Segel system, focusing on stationary solutions and convergence rates in the subcritical mass case.

Contribution

It introduces new inequalities tailored to the Keller-Segel system in self-similar variables, enhancing understanding of stationary solutions and their stability.

Findings

Derived inequalities characterize stationary solutions.

Established convergence rates towards stationary states.

Linked spectral gap inequalities to the system's dynamics.

Abstract

This note is devoted to several inequalities deduced from a special form of the logarithmic Hardy-Littlewood-Sobolev, which is well adapted to the characterization of stationary solutions of a Keller-Segel system written in self-similar variables, in case of a subcritical mass. For the corresponding evolution problem, such functional inequalities play an important role for identifying the rate of convergence of the solutions towards the stationary solution with same mass.