Stabilization in a two-dimensional attraction–repulsion Stokes system with consumption of chemoattractant

Abstract

This paper is concerned with the following attraction–repulsion Stokes system $$\begin{aligned} \left\{ \begin{array}{ll} n_t+u\cdot \nabla n=\Delta n-\chi \nabla \cdot (n\nabla c)+\xi \nabla \cdot (n\nabla v),&{}x\in \Omega ,~t>0,\\ c_t+u\cdot \nabla c=\Delta c-nc,&{}x\in \Omega ,~t>0,\\ v_t+u\cdot \nabla v=\Delta v-v+n,&{}x\in \Omega ,~t>0,\\ u_t+\nabla P=\Delta u+n\nabla \phi ,&{}x\in \Omega ,~t>0,\\ \nabla \cdot u=0,&{}x\in \Omega ,~t>0, \end{array}\right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^2$$ is a general bounded domain with smooth boundary. It is shown that for any properly regular initial data, the above system with homogeneous Neumann boundary condition admits a unique classical solution which is globally bounded and particularly, if the uniform $$L^{\infty }$$ -module of n fulfills $$\begin{aligned} \Vert n\Vert _{L^{\infty }(\Omega \times (0,\infty ))}<\frac{2}{K_{\Omega }m\xi ^2} \end{aligned}$$ with $$m:=\int _{\Omega }n_0$$ and $$K_{\Omega }>0$$ only depending on $$\Omega ,$$ then $$\begin{aligned} \Vert n(\cdot ,t)-{\bar{n}}_0\Vert _{L^{\infty }(\Omega )} +\Vert c(\cdot ,t)\Vert _{W^{1,\infty }(\Omega )}+\Vert v(\cdot ,t) -{\bar{n}}_0\Vert _{W^{1,\infty }(\Omega )}+\Vert u(\cdot ,t)\Vert _{L^{\infty }(\Omega )}\rightarrow 0\quad \text {as}\quad t\rightarrow \infty , \end{aligned}$$ where $${\bar{n}}_0:=\frac{1}{|\Omega |}\int _{\Omega }n_0.$$