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

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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, ().
[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.
[8]	A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004. doi: 10.1142/S0218202510004507.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system,, preprint, ().
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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, ().
[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.
[8]	A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004. doi: 10.1142/S0218202510004507.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system,, preprint, ().
[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.
[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.
[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.

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