Stability of a class of supercritical volume-filling chemotaxis-fluid model near Couette flow

TL;DR

This paper studies the stability of a supercritical chemotaxis-fluid model with volume-filling effects near Couette flow, demonstrating global solutions under certain initial mass and shear flow strength conditions.

Contribution

It establishes the stability and global existence of solutions for a supercritical chemotaxis-fluid system near Couette flow, under specific initial mass and shear flow strength constraints.

Findings

Solutions are globally stable if initial mass M<2π/√3.

Large enough shear flow A ensures stability.

Stability holds for supercritical volume-filling chemotaxis models.

Abstract

Consider a class of chemotaxis-fluid model incorporating a volume-filling effect in the sense of Painter and Hillen (Can. Appl. Math. Q. 2002; 10(4): 501-543), which is a supercritical parabolic-elliptic Keller-Segel system. As shown by Winkler et al., for any given mass, there exists a corresponding solution of the same mass that blows up in either finite or infinite time. In this paper, we investigate the stability properties of the two dimensional Patlak-Keller-Segel-type chemotaxis-fluid model near the Couette flow $ (Ay, 0) $ in $ \mathbb{T}\times\mathbb{R}, $ and show that the solutions are global in time as long as the initial cell mass $M<\frac{2\pi}{\sqrt{3}} $ and the shear flow is sufficiently strong ($A$ is large enough).