# Article

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Keywords:
Chemotaxis, Navier–Stokes, Lotka–Volterra, large-time behaviour
Summary:
This paper is concerned with the two-species chemotaxis-Navier–Stokes system with Lotka–Volterra competitive kinetics \begin{align*} \begin{cases} (\n1)_t+u\cdot\na\n1 =\D\n1-\chi_1\na\cdot(\n1\na c)+\mu_1\n1(1-\n1-a_1\n2) &\text{in}\ \om\times(0,\infty), \\ (\n2)_t+u\cdot\na\n2 =\D\n2-\chi_2\na\cdot(\n2\na c)+\mu_2\n2(1-a_2\n1-\n2) &\text{in}\ \om\times(0,\infty), \\ \h{6.3mm}c_t+u\cdot\na c =\D c-(\alpha\n1+\beta\n2)c &\text{in}\ \om\times(0,\infty), \\ \h{3.1mm}u_t+(u\cdot\na)u =\D u+\nabla P+(\gamma\n1+\d\n2)\na\Phi, \quad\na\cdot u=0 &\text{in}\ \om\times(0,\infty) \end{cases} \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where is a bounded domain in R3 with smooth boundary. Recently, in the 2-dimensional setting, global existence and stabilization of classical solutions to the above system were first established. However, the 3-dimensional case has not been studied: Because of difficulties in the Navier–Stokes system, we can not expect existence of classical solutions to the above system. The purpose of this paper is to obtain global existence of weak solutions to the above system, and their eventual smoothness and stabilization.
References:
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