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

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 $M_+(\overline{\Omega})$.