On the existence of global smooth solutions to the parabolic-elliptic Keller-Segel system with irregular initial data

TL;DR

This paper investigates conditions under which the parabolic-elliptic Keller-Segel system admits global smooth solutions despite irregular initial data, focusing on chemotactic attraction and repulsion in two and three dimensions.

Contribution

It establishes new criteria for the existence of global classical solutions with irregular initial data in the Keller-Segel system across different dimensions and chemotactic scenarios.

Findings

Global solutions exist for certain initial measures and parameters.

Solutions are continuous at initial time in the vague topology.

Results cover both chemotactic attraction and repulsion cases.

Abstract

We consider the parabolic-elliptic Keller-Segel system \[ \left\{ \begin{aligned} u_t &= \Delta u - \chi \nabla \cdot (u \nabla v), \\ 0 &= \Delta v - v + u \end{aligned} \right. \tag{$\star$} \] in a smooth bounded domain $\Omega \subseteq \mathbb{R}^n$, $n\in\mathbb{N}$, with Neumann boundary conditions. We look at both chemotactic attraction ($\chi > 0$) and repulsion ($\chi < 0$) scenarios in two and three dimensions. The key feature of interest for the purposes of this paper is under which conditions said system still admits global classical solutions due to the smoothing properties of the Laplacian even if the initial data is very irregular. Regarding this, we show for initial data $\mu \in \mathcal{M}_+(\overline{\Omega})$ that, if either \begin{align} \bullet\;& n = 2,\, \chi < 0\, \text{ or } \\ \bullet\;& n = 2,\, \chi > 0\, \text{ and the initial mass is small or } \\ \bullet\;& n = 3,\, \chi < 0\, \text{ and } \mu = f \in L^p(\Omega), p > 1 \end{align} holds, it is still possible to construct global classical solutions to ($\star$), which are continuous in $t = 0$ in the vague topology on $M_+(\overline{\Omega})$.