Stable singularity formation for the Keller-Segel system in three dimensions

TL;DR

This paper proves the nonlinear stability of a self-similar singularity formation in the three-dimensional Keller-Segel system, using similarity variables and spectral analysis, marking the first such result for this model.

Contribution

It establishes the nonlinear stability of an explicit self-similar blowup solution in 3D Keller-Segel, extending the understanding of singularity formation in this system.

Findings

First proof of stable self-similar blowup in Keller-Segel.

Method applicable to higher dimensions and other parabolic models.

Uses spectral analysis and semigroup theory in similarity variables.

Abstract

We consider the parabolic-elliptic Keller-Segel system in dimensions $d \geq 3$, which is the mass supercritical case. This system is known to exhibit rich dynamical behavior including singularity formation via self-similar solutions. An explicit example has been found more than two decades ago by Brenner et al. \cite{BCKSV99}, and is conjectured to be nonlinearly radially stable. We prove this conjecture for $d=3$. Our approach consists of reformulating the problem in similarity variables and studying the Cauchy evolution in intersection Sobolev spaces via semigroup theory methods. To solve the underlying spectral problem, we crucially rely on a technique we recently developed in \cite{GloSch20}. To our knowledge, this provides the first result on stable self-similar blowup for the Keller-Segel system. Furthermore, the extension of our result to any higher dimension is straightforward. We point out that our approach is general and robust, and can therefore be applied to a wide class of parabolic models.