Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane

TL;DR

This paper proves that solutions to the 2D Keller-Segel model with subcritical mass converge exponentially fast to stationary states in relative entropy, regardless of the total mass, using symmetrization and spectral gap techniques.

Contribution

It extends previous results by showing exponential convergence for all subcritical masses without restrictions, employing new spectral gap estimates and symmetrization methods.

Findings

Exponential convergence rate in relative entropy for all subcritical masses

Uniform Lp norm estimates for solutions

New spectral gap estimates for the linearized operator

Abstract

We investigate the large-time behavior of the solutions of the two-dimensional Keller-Segel system in self-similar variables, when the total mass is subcritical, that is less than 8\pi after a proper adimensionalization. It was known from previous works that all solutions converge to stationary solutions, with exponential rate when the mass is small. Here we remove this restriction and show that the rate of convergence measured in relative entropy is exponential for any mass in the subcritical range, and independent of the mass. The proof relies on symmetrization techniques, which are adapted from a paper of J.I. Diaz, T. Nagai, and J.-M. Rakotoson, and allow us to establish uniform estimates for Lp norms of the solution. Exponential convergence is obtained by the mean of a linearization in a space which is defined consistently with relative entropy estimates and in which the linearized evolution operator is self-adjoint. The core of proof relies on several new spectral gap estimates which are of independent interest.