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

Hai-Yang Jin; Tian Xiang

doi:10.3934/dcdsb.2018249

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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

Hai-Yang Jin ^1, and Tian Xiang ^2,,

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 April 2019 Early access 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

$-\Delta$

$\Omega$

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

$d$

Keywords: Chemotaxis-fluid system, boundedness, exponential convergence, algebraic convergence, convergence rates.

Mathematics Subject Classification: Primary: 35B40, 35K55, 35B44, 35K57; Secondary: 35Q92, 92C17.

Citation: Hai-Yang Jin, Tian Xiang. Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes sytstem with competitive kinetics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1919-1942. doi: 10.3934/dcdsb.2018249