Global existence and asymptotic behavior of classical solutions for a 3D two-species Keller--Segel-Stokes system with competitive kinetics

Abstract

This paper deals with the two-species Keller--Segel-Stokes system with competitive kinetics $(n_1{)}_t+u\cdot \nabla n_1=\Delta n_1-{\chi }_1\nabla \cdot (n_1\nabla c)+{\mu }_1{n}_1(1-n_1-a_1{n}_2)$, $(n_2{)}_t+u\cdot \nabla n_2=\Delta n_2-{\chi }_2\nabla \cdot (n_2\nabla c)+{\mu }_2{n}_2(1-a_2{n}_1-n_2),c_t+u\cdot \nabla c=\Delta c-c+\alpha n_1+\beta n_2$, $u_t=\Delta u+\nabla P+(\gamma n_1+\delta n_2)\nabla \varphi$, $\nabla \cdot u=0$ under homogeneous Neumann boundary conditions in a bounded domain $\Omega \subset {\mathbb{R}}^3$ with smooth boundary. Many mathematicians study chemotaxis-fluid systems and two-species chemotaxis systems with competitive kinetics. However, there are not many results on coupled two-species chemotaxis-fluid systems which have difficulties of the chemotaxis effect, the competitive kinetics and the fluid influence. Recently, in the two-species chemotaxis-Stokes system, where $-c+\alpha n_1+\beta n_2$ is replaced with $-(\alpha n_1+\beta n_2)c$ in the above system, global existence and asymptotic behavior of classical solutions were obtained in the 3-dimensional case under the condition that ${\mu }_1,{\mu }_2$ are sufficiently large. Nevertheless, the above system has not been studied yet; we cannot apply the same argument as in the previous works because of lacking the $L^{\infty }$-information of $c$. The main purpose of this paper is to obtain global existence and stabilization of classical solutions to the above system in the 3-dimensional case under the largeness conditions for ${\mu }_1,{\mu }_2$.

17 pages