American Institute of Mathematical Sciences

Advanced
October  2015, 20(8): 2751-2759. doi: 10.3934/dcdsb.2015.20.2751

Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes system

Qingshan Zhang 1, and  Yuxiang Li 1,

1. 

Department of Mathematics, Southeast University, Nanjing 211189, China, China

Received  November 2014 Revised  February 2015 Published  August 2015

We consider an initial-boundary value problem for the incompressible chemotaxis-Navier-Stokes equation \begin{eqnarray*} \left\{\begin{array}{lll} n_t + u \cdot \nabla n = \Delta n - \chi\nabla\cdot(n \nabla c),&{} x\in\Omega,\ t>0,\\ c_t + u \cdot \nabla c = \Delta c - nc, &{} x \in \Omega,\ t>0,\\ u_t + \kappa(u\cdot\nabla)u = \Delta u + \nabla P + n\nabla\phi ,&{} x\in\Omega,\ t>0,\\ \nabla\cdot u=0, &{}x\in\Omega,\ t>0, \end{array}\right. \end{eqnarray*} in a bounded domain $\Omega\subset\mathbb{R}^2$. It is known that if $\chi>0$, $\kappa\in\mathbb{R}$ and $\phi\in C^2(\bar{\Omega})$, for sufficiently smooth initial data, the model possesses a unique global classical solution which satisfies $(n, c, u)\rightarrow(\bar{n}_0, 0, 0)$ as $t\rightarrow\infty$ uniformly with respect to $x\in\Omega$, where $\bar{n}_0:=\frac{1}{|\Omega|}\int_{\Omega}n(x, 0)dx$. In the present paper, we prove this solution converges to $(\bar{n}_0, 0, 0)$ exponentially in time.
Keywords: asymptotic behavior, Chemotaxis-Navier-Stokes equation, convergence rate..
Mathematics Subject Classification: Primary: 35K55, 35Q35; Secondary: 41A25, 92C1.
Citation: 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
References:
[1]

M. Chae, K. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297. doi: 10.3934/dcds.2013.33.2271.  Google Scholar

[2]

M. Chae, K. Kang and J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differential Equations, 39 (2014), 1205-1235. doi: 10.1080/03605302.2013.852224.  Google Scholar

[3]

M. Di Francesco, A. Lorz and P. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., 28 (2010), 1437-1453. doi: 10.3934/dcds.2010.28.1437.  Google Scholar

[4]

R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673. doi: 10.1080/03605302.2010.497199.  Google Scholar

[5]

R. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Not. IMRN, (2014), 1833-1852.  Google Scholar

[6]

J. Jiang, H. Wu and S. Zheng, Global existence and asymptotic behavior of solutions to a chemotaxis-fluid system on general bounded domain,, preprint, ().   Google Scholar

[7]

J.-G. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652. doi: 10.1016/j.anihpc.2011.04.005.  Google Scholar

[8]

A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004. doi: 10.1142/S0218202510004507.  Google Scholar

[9]

A. Lorz, A coupled Keller-Segel-Stokes model: Global existence for small initial data and blow-up delay, Commun. Math. Sci., 10 (2012), 555-574. doi: 10.4310/CMS.2012.v10.n2.a7.  Google Scholar

[10]

Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 1901-1914. doi: 10.3934/dcds.2012.32.1901.  Google Scholar

[11]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178. doi: 10.1016/j.anihpc.2012.07.002.  Google Scholar

[12]

I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proceedings of the National Academy of Sciences of the United States of America, 102 (2005), 2277-2282. doi: 10.1073/pnas.0406724102.  Google Scholar

[13]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[14]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351. doi: 10.1080/03605302.2011.591865.  Google Scholar

[15]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077. doi: 10.1016/j.jde.2014.04.023.  Google Scholar

[16]

M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system,, preprint, ().   Google Scholar

[17]

M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487. doi: 10.1007/s00205-013-0678-9.  Google Scholar

[18]

Q. Zhang, Local well-posedness for the chemotaxis-Navier-Stokes equations in Besov spaces, Nonlinear Anal. Real World Appl., 17 (2014), 89-100. doi: 10.1016/j.nonrwa.2013.10.008.  Google Scholar

[19]

Q. Zhang and X. Zheng, Global well-posedness for the two-dimensional incompressible chemotaxis-Navier-Stokes equations, SIAM J. Math. Anal., 46 (2014), 3078-3105. doi: 10.1137/130936920.  Google Scholar

show all references

References:
[1]

M. Chae, K. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297. doi: 10.3934/dcds.2013.33.2271.  Google Scholar

[2]

M. Chae, K. Kang and J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differential Equations, 39 (2014), 1205-1235. doi: 10.1080/03605302.2013.852224.  Google Scholar

[3]

M. Di Francesco, A. Lorz and P. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., 28 (2010), 1437-1453. doi: 10.3934/dcds.2010.28.1437.  Google Scholar

[4]

R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673. doi: 10.1080/03605302.2010.497199.  Google Scholar

[5]

R. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Not. IMRN, (2014), 1833-1852.  Google Scholar

[6]

J. Jiang, H. Wu and S. Zheng, Global existence and asymptotic behavior of solutions to a chemotaxis-fluid system on general bounded domain,, preprint, ().   Google Scholar

[7]

J.-G. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652. doi: 10.1016/j.anihpc.2011.04.005.  Google Scholar

[8]

A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004. doi: 10.1142/S0218202510004507.  Google Scholar

[9]

A. Lorz, A coupled Keller-Segel-Stokes model: Global existence for small initial data and blow-up delay, Commun. Math. Sci., 10 (2012), 555-574. doi: 10.4310/CMS.2012.v10.n2.a7.  Google Scholar

[10]

Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 1901-1914. doi: 10.3934/dcds.2012.32.1901.  Google Scholar

[11]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178. doi: 10.1016/j.anihpc.2012.07.002.  Google Scholar

[12]

I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proceedings of the National Academy of Sciences of the United States of America, 102 (2005), 2277-2282. doi: 10.1073/pnas.0406724102.  Google Scholar

[13]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[14]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351. doi: 10.1080/03605302.2011.591865.  Google Scholar

[15]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077. doi: 10.1016/j.jde.2014.04.023.  Google Scholar

[16]

M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system,, preprint, ().   Google Scholar

[17]

M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487. doi: 10.1007/s00205-013-0678-9.  Google Scholar

[18]

Q. Zhang, Local well-posedness for the chemotaxis-Navier-Stokes equations in Besov spaces, Nonlinear Anal. Real World Appl., 17 (2014), 89-100. doi: 10.1016/j.nonrwa.2013.10.008.  Google Scholar

[19]

Q. Zhang and X. Zheng, Global well-posedness for the two-dimensional incompressible chemotaxis-Navier-Stokes equations, SIAM J. Math. Anal., 46 (2014), 3078-3105. doi: 10.1137/130936920.  Google Scholar

[1]

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
[2]

Minghua Yang, Zunwei Fu, Jinyi Sun. Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3427-3460. doi: 10.3934/dcdsb.2018284
[3]

Mimi Dai, Han Liu. Low modes regularity criterion for a chemotaxis-Navier-Stokes system. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2713-2735. doi: 10.3934/cpaa.2020118
[4]

Laiqing Meng, Jia Yuan, Xiaoxin Zheng. Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3413-3441. doi: 10.3934/dcds.2019141
[5]

Xiaoping Zhai, Zhaoyang Yin. Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2829-2859. doi: 10.3934/dcds.2017122
[6]

Yulan Wang. Global solvability in a two-dimensional self-consistent chemotaxis-Navier-Stokes system. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 329-349. doi: 10.3934/dcdss.2020019
[7]

Changchun Liu, Pingping Li. Time periodic solutions for a two-species chemotaxis-Navier-Stokes system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4567-4585. doi: 10.3934/dcdsb.2020303
[8]

Jiayi Han, Changchun Liu. Global existence for a two-species chemotaxis-Navier-Stokes system with $ p $-Laplacian. Electronic Research Archive, , () : -. doi: 10.3934/era.2021050
[9]

Boris Haspot, Ewelina Zatorska. From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3107-3123. doi: 10.3934/dcds.2016.36.3107
[10]

Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907
[11]

Sachiko Ishida. Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3463-3482. doi: 10.3934/dcds.2015.35.3463
[12]

Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073
[13]

Zhong Tan, Xu Zhang, Huaqiao Wang. Asymptotic behavior of Navier-Stokes-Korteweg with friction in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2243-2259. doi: 10.3934/dcds.2014.34.2243
[14]

Zhong Tan, Leilei Tong. Asymptotic behavior of the compressible non-isentropic Navier-Stokes-Maxwell system in $\mathbb{R}^3$. Kinetic & Related Models, 2018, 11 (1) : 191-213. doi: 10.3934/krm.2018010
[15]

Bo-Qing Dong, Juan Song. Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 57-79. doi: 10.3934/dcds.2012.32.57
[16]

Changjiang Zhu, Ruizhao Zi. Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1263-1283. doi: 10.3934/dcds.2011.30.1263
[17]

Tomás Caraballo, Xiaoying Han. A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1079-1101. doi: 10.3934/dcdss.2015.8.1079
[18]

Anhui Gu, Kening Lu, Bixiang Wang. Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 185-218. doi: 10.3934/dcds.2019008
[19]

G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123
[20]

Xinhua Zhao, Zilai Li. Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1421-1448. doi: 10.3934/cpaa.2020052

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (161)
  • HTML views (0)
  • Cited by (53)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences