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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  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, 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

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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

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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

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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

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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

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