# American Institute of Mathematical Sciences

• Previous Article
Lyapunov type inequalities for Hammerstein integral equations and applications to population dynamics
• DCDS-B Home
• This Issue
• Next Article
Asymptotic behavior for stochastic plate equations with rotational inertia and Kelvin-Voigt dissipative term on unbounded domains
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:

show all references

##### References:
 [1] Qingshan Zhang, Yuxiang Li. Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2751-2759. doi: 10.3934/dcdsb.2015.20.2751 [2] James Broda, Alexander Grigo, Nikola P. Petrov. Convergence rates for semistochastic processes. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 109-125. doi: 10.3934/dcdsb.2019001 [3] Guoqiang Ren, Bin Liu. Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3843-3883. doi: 10.3934/cpaa.2020170 [4] Mina Jiang, Changjiang Zhu. Convergence rates to nonlinear diffusion waves for $p$-system with nonlinear damping on quadrant. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 887-918. doi: 10.3934/dcds.2009.23.887 [5] Masashi Ohnawa. Convergence rates towards the traveling waves for a model system of radiating gas with discontinuities. Kinetic & Related Models, 2012, 5 (4) : 857-872. doi: 10.3934/krm.2012.5.857 [6] Hongyun Peng, Lizhi Ruan, Changjiang Zhu. Convergence rates of zero diffusion limit on large amplitude solution to a conservation laws arising in chemotaxis. Kinetic & Related Models, 2012, 5 (3) : 563-581. doi: 10.3934/krm.2012.5.563 [7] Frank Blume. Minimal rates of entropy convergence for rank one systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 773-796. doi: 10.3934/dcds.2000.6.773 [8] Jie Zhao. Convergence rates for elliptic reiterated homogenization problems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2787-2795. doi: 10.3934/cpaa.2013.12.2787 [9] Wilhelm Schlag. Regularity and convergence rates for the Lyapunov exponents of linear cocycles. Journal of Modern Dynamics, 2013, 7 (4) : 619-637. doi: 10.3934/jmd.2013.7.619 [10] Stefan Kindermann, Antonio Leitão. Convergence rates for Kaczmarz-type regularization methods. Inverse Problems & Imaging, 2014, 8 (1) : 149-172. doi: 10.3934/ipi.2014.8.149 [11] Feng-Yu Wang. Exponential convergence of non-linear monotone SPDEs. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5239-5253. doi: 10.3934/dcds.2015.35.5239 [12] Jonathan Zinsl. Exponential convergence to equilibrium in a Poisson-Nernst-Planck-type system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2915-2930. doi: 10.3934/dcds.2016.36.2915 [13] Narcisse Batangouna, Morgan Pierre. Convergence of exponential attractors for a time splitting approximation of the Caginalp phase-field system. Communications on Pure & Applied Analysis, 2018, 17 (1) : 1-19. doi: 10.3934/cpaa.2018001 [14] Weisheng Niu, Yao Xu. Convergence rates in homogenization of higher-order parabolic systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4203-4229. doi: 10.3934/dcds.2018183 [15] Zhong Tan, Qiuju Xu, Huaqiao Wang. Global existence and convergence rates for the compressible magnetohydrodynamic equations without heat conductivity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5083-5105. doi: 10.3934/dcds.2015.35.5083 [16] Daniel Gerth, Andreas Hofinger, Ronny Ramlau. On the lifting of deterministic convergence rates for inverse problems with stochastic noise. Inverse Problems & Imaging, 2017, 11 (4) : 663-687. doi: 10.3934/ipi.2017031 [17] Stefano Galatolo, Isaia Nisoli, Benoît Saussol. An elementary way to rigorously estimate convergence to equilibrium and escape rates. Journal of Computational Dynamics, 2015, 2 (1) : 51-64. doi: 10.3934/jcd.2015.2.51 [18] L. Olsen. Rates of convergence towards the boundary of a self-similar set. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 799-811. doi: 10.3934/dcds.2007.19.799 [19] José A. Carrillo, Jean Dolbeault, Ivan Gentil, Ansgar Jüngel. Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1027-1050. doi: 10.3934/dcdsb.2006.6.1027 [20] Xiuli Xu, Xueke Pu. Optimal convergence rates of the magnetohydrodynamic model for quantum plasmas with potential force. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020150

2018 Impact Factor: 1.008