Critical Keller-Segel meets Burgers on ${\mathbb S}^1$: large-time smooth solutions

TL;DR

This paper proves that solutions to the critical Keller-Segel system on the circle remain smooth for all time, challenging previous blow-up conjectures, and introduces a novel approach via a generalized modulus of continuity.

Contribution

It demonstrates global smoothness of the critical Keller-Segel system on ${ m S}^1$, and develops a new method for analyzing a modified Burgers equation without scaling.

Findings

Solutions remain smooth for all initial data and positive times.

Disproves the large-data-blowup conjecture in the periodic setting.

Provides improved understanding of the asymptotic behavior of solutions.

Abstract

We show that solutions to the parabolic-elliptic Keller-Segel system on ${\mathbb S}^1$ with critical fractional diffusion $(-\Delta)^\frac{1}{2}$ remain smooth for any initial data and any positive time. This disproves, at least in the periodic setting, the large-data-blowup conjecture by Bournaveas and Calvez. As a tool, we show smoothness of solutions to a modified critical Burgers equation via a generalization of the method of moduli of continuity by Kiselev, Nazarov and Shterenberg. over a setting where the considered equation has no scaling. This auxiliary result may be interesting by itself. Finally, we study the asymptotic behavior of global solutions, improving the existing results.