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April  2019, 24(4): 1919-1942. doi: 10.3934/dcdsb.2018249

## Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes sytstem with competitive kinetics

 1 Department of Mathematics, South China University of Technology, Guangzhou 510640, China 2 Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China

* Corresponding author

Received  January 2018 Revised  April 2018 Published  August 2018

We study the convergence rates of solutions to the two-species chemotaxis-Navier-Stokes system with Lotka-Volterra competitive kinetics:
 $\begin{equation*} \begin{cases} & (n_1)_t + u\cdot\nabla n_1 = \Delta n_1 - \chi_1\nabla\cdot(n_1\nabla c) + \mu_1n_1(1- n_1 - a_1n_2), \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in \Omega,\ t>0, \\ & (n_2)_t + u\cdot\nabla n_2 = \Delta n_2 - \chi_2\nabla\cdot(n_2\nabla c) + \mu_2n_2(1- a_2n_1 - n_2), \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in \Omega,\ t>0, \\ & c_t + u\cdot\nabla c = \Delta c -(\alpha n_1 + \beta n_2)c, x \in \Omega,\ t>0, \\ & \ u_t + \kappa (u\cdot\nabla) u = \Delta u + \nabla P + (\gamma n_1 + \delta n_2)\nabla\phi, \quad \nabla\cdot u = 0, \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x \in \Omega,\ t>0 \end{cases} \end{equation*}$
under homogeneous Neumann boundary conditions for
 $n_1,n_2,c$
and no-slip boundary condition for
 $u$
in a bounded domain
 $\Omega \subset \mathbb{R}^d(d\in\{2,3\})$
with smooth boundary. The global existence, boundedness and stabilization of solutions have been obtained in
 $2$
-D [8] and
 $3$
-D for
 $\kappa = 0$
and
 $\frac{\max\{\chi_1,\chi_2\}}{\min\{\mu_1,\mu_2\}}\|c_0\|_{L^\infty(\Omega)}$
being sufficiently small [4]. Here, we examine further convergence and derive the explicit rates of convergence for any supposedly given global bounded classical solution
 $(n_1, n_2, c, u)$
; more specifically, in
 $L^\infty$
-topology, we show that
 $(n_1(\cdot,t), n_2(\cdot,t), u(\cdot,t))\overset{t\rightarrow\infty}\rightarrow \begin{cases} (\frac{1 - a_1}{1 - a_1a_2},\frac{1 - a_2}{1 - a_1a_2},0) \text{ exponentially,}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{ if } a_1, a_2 \in (0, 1), \\ (0,1,0) \text{ exponentially, if } a_1>1> a_2, \\ (0,1,0) \text{ algebraically, if } a_1 = 1> a_2, \\ (1,,0,0) \text{ exponentially, if } a_2>1> a_1, \\ (1,0,0) \text{ algebraically, if } a_2 = 1> a_1. \end{cases}$
In either cases, the
 $c$
-solution component converges exponentially to
 $0$
.
Moreover, it is shown that only the rate of convergence for
 $u$
is expressed in terms of the model parameters and the first eigenvalue of
 $-\Delta$
in
 $\Omega$
under homogeneous Dirichlet boundary conditions, and all other rates of convergence are explicitly expressed only in terms of the model parameters
 $a_i, \mu_i, \alpha$
and
 $\beta$
and the space dimension
 $d$
.
Citation: Hai-Yang Jin, Tian Xiang. Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes sytstem with competitive kinetics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1919-1942. doi: 10.3934/dcdsb.2018249
##### References:

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