Does strong repulsion lead to smooth solutions in a repulsion-attraction chemotaxis system even when starting with highly irregular initial data?

TL;DR

This paper investigates whether strong repulsive chemotaxis can regularize solutions in a Keller-Segel system starting from highly irregular initial data, demonstrating existence of smooth solutions in certain dimensions.

Contribution

It extends previous results by establishing the existence of smooth solutions from irregular initial data under strong repulsion in 2D and 3D chemotaxis models.

Findings

Strong repulsion can lead to regular solutions even from highly irregular initial data.

Existence of smooth solutions is proven in 2D parabolic-parabolic and 2D/3D parabolic-elliptic systems.

Sufficiently strong repulsion prevents finite-time blow-up under certain conditions.

Abstract

It has been well established that, in attraction-repulsion Keller-Segel systems of the form\begin{equation*} \left\{ \begin{aligned} u_t &= \Delta u - \chi \nabla \cdot (u\nabla v) + \xi \nabla \cdot (u\nabla w), \\ \tau v_t &= \Delta v + \alpha u - \beta v,\\ \tau w_t &= \Delta w + \gamma u - \delta w \end{aligned} \right. \end{equation*} in a smooth bounded domain $\Omega \subseteq \mathbb{R}^n$, $n\in\mathbb{N}$, with Neumann boundary conditions and parameters $\chi, \xi \geq 0$, $\alpha,\beta,\gamma,\delta > 0$ and $\tau \in \{0,1\}$, finite-time blow-up can be ruled out in many scenarios given sufficiently smooth initial data if the repulsive chemotaxis is sufficiently stronger than its attractive counterpart. In this paper, we will go - in a sense - a step further than this by studying the same system with initial data that could already be understood as being in a blown-up state (e.g. a positive Radon measure for the first solution component) and then ask the question whether sufficiently strong repulsion has enough of a regularizing effect to lead to the existence of a smooth solution, which is still connected to said initial data in a sensible fashion. Regarding this, we in fact establish that the construction of such a solution is possible in the two-dimensional parabolic-parabolic system and the two- and three-dimensional parabolic-elliptic system under appropriate assumptions on the interaction of repulsion and attraction as well as the initial data.