Uniqueness and long time asymptotics for the parabolic-parabolic Keller-Segel equation

TL;DR

This paper proves the uniqueness of weak solutions for the parabolic-parabolic Keller-Segel equation with subcritical mass and demonstrates exponential stability of self-similar profiles in the radially symmetric case.

Contribution

It establishes the first proof of uniqueness for weak solutions with finite mass less than 8π and finite entropy, and shows exponential stability of self-similar profiles in the quasi parabolic-elliptic regime.

Findings

Uniqueness of free energy solutions for M<8π.

Exponential stability of self-similar profiles in the radially symmetric case.

Application of DiPerna-Lions renormalization and perturbation methods.

Abstract

The present paper deals with the parabolic-parabolic Keller-Segel equation in the plane inthe general framework of weak (or "free energy") solutions associated to an initial datum with finite mass $M\textless{} 8\pi$, finite second log-moment and finite entropy. The aim of the paper is twofold:(1) We prove the uniqueness of the "free energy" solution. The proof uses a DiPerna-Lions renormalizing argument which makes possible to get the "optimal regularity" as well as an estimate of the difference of two possible solutions in the critical $L^{4/3}$ Lebesgue norm similarly as for the $2d$ vorticity Navier-Stokes equation. (2) We prove a radially symmetric and polynomial weighted $L^2$ exponential stability of the self-similar profile in the quasi parabolic-elliptic regime. The proof is based on a perturbation argument which takes advantage of the exponential stability of the self-similar profile for the parabolic-elliptic Keller-Segel equation established by Campos-Dolbeault and Egana-Mischler.