Refined Asymptotics for the subcritical Keller-Segel system and Related Functional Inequalities

TL;DR

This paper investigates the convergence rate to self-similarity in the subcritical Keller-Segel system, providing refined asymptotics, and establishes related functional inequalities using optimal transport and Fourier analysis.

Contribution

It offers new asymptotic results for the Keller-Segel system and proves a logarithmic Hardy-Littlewood-Sobolev inequality via optimal transport methods.

Findings

Convergence rate remains stable near the critical case.

Proof of the logarithmic Hardy-Littlewood-Sobolev inequality in 1D and radially symmetric 2D cases.

The 1D equation exhibits contraction in Fourier distance in the subcritical regime.

Abstract

We analyze the rate of convergence towards self-similarity for the subcritical Keller-Segel system in the radially symmetric two-dimensional case and in the corresponding one-dimensional case for logarithmic interaction. We measure convergence in Wasserstein distance. The rate of convergence towards self-similarity does not degenerate as we approach the critical case. As a byproduct, we obtain a proof of the logarithmic Hardy-Littlewood-Sobolev inequality in the one dimensional and radially symmetric two dimensional case based on optimal transport arguments. In addition we prove that the one-dimensional equation is a contraction with respect to Fourier distance in the subcritical case.