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}&=Δn-\nabla\!\cdot(n\nabla c)-u\cdot\!\nabla n,\ &x\inΩ,& t>0,\\ c_{t}&=Δc-c+f(n)-u\cdot\!\nabla c,\ &x\inΩ,& t>0,\\ u_{t}&=Δu+\nabla P+n\cdot\!\nablaϕ,\ &x\inΩ,& t>0,\\ \nabla\cdot u&=0,\ &x\inΩ,& t>0, \end{array}\right. \end{align*} where $\Omega \subset {\mathbb{R}}^2$ is a bounded and convex domain with smooth boundary, $\varphi \in W^{1,\infty }\left(\Omega \right)$ and $f\in C^1([0,\infty ))$ satisfies $0\le f(s)\le K_0{s}^{\alpha }$ 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.

20 pages

References in corpus (1)

• Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion

Cited by in corpus (2)

• Global existence and asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics
• Long-term behaviour in a chemotaxis-fluid system with logistic source