American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds
  • DCDS-B Home
  • This Issue
  • Next Article
    The effect of caputo fractional difference operator on a novel game theory model
August  2021, 26(8): 4567-4585. doi: 10.3934/dcdsb.2020303

Time periodic solutions for a two-species chemotaxis-Navier-Stokes system

Changchun Liu , and  Pingping Li

Department of Mathematics, Jilin University, Changchun 130012, China

* Corresponding author: Changchun Liu

Received  June 2020 Revised  August 2020 Published  August 2021 Early access  October 2020

We consider a chemotaxis-Navier-Stokes system in two dimensional bounded domains. It is asserted that the chemotaxis system admits a time periodic solution under some conditions.

Keywords: Chemotaxis system, chemotaxis-Navier-Stokes, periodic solution, existence, classical solution.
Mathematics Subject Classification: Primary: 92C17, 35B10; Secondary: 35K45, 35Q35.
Citation: 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
References:
[1]

X. Cao, Global classical solutions in chemotaxis-Navier-Stokes system with rotational flux term, J. Differential Equations, 261 (2016), 6883-6914.  doi: 10.1016/j.jde.2016.09.007.  Google Scholar

[2]

X. Cao, S. Kurima and M. Mizukami, Global existence and asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics, Math. Meth. Appl. Sci., 41 (2018), 3138-3154.  doi: 10.1002/mma.4807.  Google Scholar

[3]

C. Jin, Large time periodic solutions to coupled chemotaxis-fluid models, Z. Angew. Math. Phys., 68 (2017), 24 pp. doi: 10.1007/s00033-017-0882-9.  Google Scholar

[4]

C. Jin, Large time periodic solution to the coupled chemotaxis-Stokes model, Math. Nachr., 290 (2017), 1701-1715.  doi: 10.1002/mana.201600180.  Google Scholar

[5]

H. Jin and T. Xiang, Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes system with competitive kinetics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1919-1942.  doi: 10.3934/dcdsb.2018249.  Google Scholar

[6]

R. Farwig and T. Okabe, Periodic solutions of the Navier-Stokes equations with inhomogeneous boundary conditions, Ann. Univ. Ferrara Sez. VII Sci. Mat., 56 (2010), 249-281.  doi: 10.1007/s11565-010-0108-y.  Google Scholar

[7]

J. Liu, Boundedness in a Chemotaxis-Navier-Stokes system modeling coral fertilization with slow p-Laplacian diffusion, J. Math. Fluid Mech., 22 (2020), 31 pp. doi: 10.1007/s00021-019-0469-7.  Google Scholar

[8]

W. Tao and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with slow p-Laplacian diffusion, Nonlinear Anal. Real World Appl., 45 (2019), 26-52.  doi: 10.1016/j.nonrwa.2018.06.005.  Google Scholar

[9]

W. Tao and Y. Li, Boundedness of weak solutions of a chemotaxis-Stokes system with slow p-Laplacian diffusion, J. Differential Equations, 268 (2020), 6872-6919.  doi: 10.1016/j.jde.2019.11.078.  Google Scholar

[10]

I. Tuval, L. Cisneros, C. Dombrowski, C. Wolgemuth, J. Kessler and R. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102.  Google Scholar

[11]

Y. Wang, M. Winkler and Z. Xiang, Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 18 (2018), 421-466.  doi: 10.2422/2036-2145.201603_004.  Google Scholar

[12]

M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.  Google Scholar

[13]

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

[14]

M. Winkler, How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans. Amer. Math. Soc., 369 (2017), 3067-3125.  doi: 10.1090/tran/6733.  Google Scholar

[15]

Q. Zhang and Y. Li, Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes system, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2751-2759.  doi: 10.3934/dcdsb.2015.20.2751.  Google Scholar

[16]

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]

X. Cao, Global classical solutions in chemotaxis-Navier-Stokes system with rotational flux term, J. Differential Equations, 261 (2016), 6883-6914.  doi: 10.1016/j.jde.2016.09.007.  Google Scholar

[2]

X. Cao, S. Kurima and M. Mizukami, Global existence and asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics, Math. Meth. Appl. Sci., 41 (2018), 3138-3154.  doi: 10.1002/mma.4807.  Google Scholar

[3]

C. Jin, Large time periodic solutions to coupled chemotaxis-fluid models, Z. Angew. Math. Phys., 68 (2017), 24 pp. doi: 10.1007/s00033-017-0882-9.  Google Scholar

[4]

C. Jin, Large time periodic solution to the coupled chemotaxis-Stokes model, Math. Nachr., 290 (2017), 1701-1715.  doi: 10.1002/mana.201600180.  Google Scholar

[5]

H. Jin and T. Xiang, Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes system with competitive kinetics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1919-1942.  doi: 10.3934/dcdsb.2018249.  Google Scholar

[6]

R. Farwig and T. Okabe, Periodic solutions of the Navier-Stokes equations with inhomogeneous boundary conditions, Ann. Univ. Ferrara Sez. VII Sci. Mat., 56 (2010), 249-281.  doi: 10.1007/s11565-010-0108-y.  Google Scholar

[7]

J. Liu, Boundedness in a Chemotaxis-Navier-Stokes system modeling coral fertilization with slow p-Laplacian diffusion, J. Math. Fluid Mech., 22 (2020), 31 pp. doi: 10.1007/s00021-019-0469-7.  Google Scholar

[8]

W. Tao and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with slow p-Laplacian diffusion, Nonlinear Anal. Real World Appl., 45 (2019), 26-52.  doi: 10.1016/j.nonrwa.2018.06.005.  Google Scholar

[9]

W. Tao and Y. Li, Boundedness of weak solutions of a chemotaxis-Stokes system with slow p-Laplacian diffusion, J. Differential Equations, 268 (2020), 6872-6919.  doi: 10.1016/j.jde.2019.11.078.  Google Scholar

[10]

I. Tuval, L. Cisneros, C. Dombrowski, C. Wolgemuth, J. Kessler and R. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102.  Google Scholar

[11]

Y. Wang, M. Winkler and Z. Xiang, Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 18 (2018), 421-466.  doi: 10.2422/2036-2145.201603_004.  Google Scholar

[12]

M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.  Google Scholar

[13]

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

[14]

M. Winkler, How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans. Amer. Math. Soc., 369 (2017), 3067-3125.  doi: 10.1090/tran/6733.  Google Scholar

[15]

Q. Zhang and Y. Li, Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes system, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2751-2759.  doi: 10.3934/dcdsb.2015.20.2751.  Google Scholar

[16]

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]

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
[2]

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
[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]

Xi Wang, Zuhan Liu, Ling Zhou. Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4003-4020. doi: 10.3934/dcdsb.2018121
[5]

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
[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]

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
[8]

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
[9]

Chengxin Du, Changchun Liu. Time periodic solution to a two-species chemotaxis-Stokes system with $ p $-Laplacian diffusion. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021162
[10]

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
[11]

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
[12]

Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064
[13]

Tong Li, Anthony Suen. Existence of intermediate weak solution to the equations of multi-dimensional chemotaxis systems. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 861-875. doi: 10.3934/dcds.2016.36.861
[14]

Jie Zhao. Large time behavior of solution to quasilinear chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1737-1755. doi: 10.3934/dcds.2020091
[15]

Chunhua Jin. Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3547-3566. doi: 10.3934/dcds.2018150
[16]

Yulan Wang, Xinru Cao. Global classical solutions of a 3D chemotaxis-Stokes system with rotation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3235-3254. doi: 10.3934/dcdsb.2015.20.3235
[17]

Atanas Stefanov. On the Lipschitzness of the solution map for the 2 D Navier-Stokes system. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1471-1490. doi: 10.3934/dcds.2010.26.1471
[18]

Jiapeng Huang, Chunhua Jin. Time periodic solution to a coupled chemotaxis-fluid model with porous medium diffusion. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5415-5439. doi: 10.3934/dcds.2020233
[19]

Ling Liu, Jiashan Zheng. Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3357-3377. doi: 10.3934/dcdsb.2018324
[20]

Xiangdong Zhao. Global boundedness of classical solutions to a logistic chemotaxis system with singular sensitivity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 5095-5100. doi: 10.3934/dcdsb.2020334

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (169)
  • HTML views (330)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences