Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller-Segel equation

TL;DR

This paper establishes new stability results for Gagliardo-Nirenberg-Sobolev and Log-HLS inequalities in the plane, and applies these to prove quantitative convergence for the critical mass Keller-Segel system.

Contribution

It extends stability analysis from the 2-Sobolev inequality to Gagliardo-Nirenberg-Sobolev and Log-HLS inequalities, and applies these to the Keller-Segel equation.

Findings

Quantitative stability results for Gagliardo-Nirenberg-Sobolev inequality.

Stability results for Log-HLS inequality derived from Gagliardo-Nirenberg-Sobolev stability.

Quantitative convergence for the critical mass Keller-Segel system.

Abstract

Starting from the quantitative stability result of Bianchi and Egnell for the 2-Sobolev inequality, we deduce several different stability results for a Gagliardo-Nirenberg-Sobolev inequality in the plane. Then, exploiting the connection between this inequality and a fast diffusion equation, we get a quantitative stability for the Log-HLS inequality. Finally, using all these estimates, we prove a quantitative convergence result for the critical mass Keller-Segel system.