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

TL;DR

This paper proves that in a two-dimensional chemotaxis-fluid system with sublinear signal production, solutions exist globally and stay bounded over time, extending understanding of cell movement in fluid environments.

Contribution

It establishes the global existence and boundedness of classical solutions for a Keller-Segel-Stokes system with sublinear signal production in two dimensions.

Findings

Solutions are global for all time in 2D when 0<α<1.

Solutions remain bounded over time.

The system models chemotactic cell movement in fluid environments.

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.