Spectral instability and non-uniqueness of mild solutions for the Keller-Segel system

TL;DR

This paper demonstrates the local ill-posedness and spectral instability of solutions to the Keller-Segel system in certain function spaces, highlighting non-uniqueness driven by self-similarity instabilities.

Contribution

It establishes spectral instability and non-uniqueness of mild solutions for the Keller-Segel model in supercritical spaces across multiple dimensions.

Findings

The Keller-Segel problem is locally ill-posed in L^q spaces for dimensions 3 to 9.

Non-uniqueness of solutions is caused by an instability mechanism in self-similarity variables.

The instability mechanism is similar to that proposed for the Navier-Stokes equations.

Abstract

We show that the Cauchy problem associated with the parabolic-elliptic Keller-Segel model is locally ill-posed in $L^q(\mathbb{R}^n)$ for dimensions $n \in \{3,\dots,9\}$ and throughout the supercritical range $q\in [1,\frac{n}{2})$. The non-uniqueness is driven by an instability mechanism in self-similarity variables, in the spirit of the program proposed by Jia and \v{S}ver\'ak for the three-dimensional Navier-Stokes equations.