Sublinear signal production in a two-dimensional Keller-Segel-Stokes system

Abstract: We study the chemotaxis-fluid system \begin{align*} \left{\begin{array}{r@{\,}l@{\quad}l@{\,}c} n_{t}&=\Delta n-\nabla!\cdot(n\nabla c)-u\cdot!\nabla n,\ &x\in\Omega,& t>0,\ c_{t}&=\Delta c-c+f(n)-u\cdot!\nabla c,\ &x\in\Omega,& t>0,\ u_{t}&=\Delta u+\nabla P+n\cdot!\nabla\phi,\ &x\in\Omega,& t>0,\ \nabla\cdot u&=0,\ &x\in\Omega,& t>0, \end{array}\right. \end{align*} where $\Omega\subset\mathbb{R}^2$ is a bounded and convex domain with smooth boundary, $\phi\in W^{1,\infty}\left(\Omega\right)$ and $f\in C^1([0,\infty))$ satisfies $0\leq f(s)\leq K_0 s^\alpha$ for all $s\in[0,\infty)$, with $K_0>0$ and $\alpha\in(0,1]$. This system models the chemotactic movement of actively communicating cells in slow moving liquid. We will show that in the two-dimensional setting for any $\alpha\in(0,1)$ the classical solution to this Keller-Segel-Stokes-system is global and remains bounded for all times.