November  2020, 40(11): 6379-6409. doi: 10.3934/dcds.2020284

Global existence and large time behavior for the chemotaxis–shallow water system in a bounded domain

1. 

School of Mathematical Sciences and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

2. 

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Yucheng Wang

Received  January 2020 Revised  June 2020 Published  July 2020

Fund Project: The authors are supported by the National Natural Science Foundation of China (No. 11771284 and No. 11831011)

In this paper, we consider the chemotaxis–shallow water system in a bounded domain $ \Omega\subset\mathbb{R}^2 $. By energy method, we establish the global existence of strong solution with small initial perturbation and obtain the exponential decaying rate of the solution. We divide the bounded domain into interior domain and the domain up to the boundary. In the interior domain, the problem is treated like the Cauchy problem. In the domain up to the boundary, the tangential and normal directions are treated differently. We use different method to get the estimates for the tangential and normal directions.

Citation: Weike Wang, Yucheng Wang. Global existence and large time behavior for the chemotaxis–shallow water system in a bounded domain. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6379-6409. doi: 10.3934/dcds.2020284
References:
[1]

X. Cao, Global bounded solutions of the higher-dimensional Keller–Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891.

[2]

X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier–Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differential Equations, 55 (2016), 39pp. doi: 10.1007/s00526-016-1027-2.

[3]

J. CheL. ChenB. Duan and Z. Luo, On the existence of local strong solutions to chemotaxis–shallow water system with large data and vacuum, J. Differential Equations, 261 (2016), 6758-6789.  doi: 10.1016/j.jde.2016.09.005.

[4]

Y. Cho and H. Kim, On classical solutions of the compressible Navier–Stokes equations with nonnegative initial densities, Manuscripta Math., 120 (2006), 91-129.  doi: 10.1007/s00229-006-0637-y.

[5]

R. DuanX. Li and Z. Xiang, Global existence and large time behavior for a two-dimensional chemotaxis–Navier–Stokes system, J. Differential Equations, 263 (2017), 6284-6316.  doi: 10.1016/j.jde.2017.07.015.

[6]

A. Duarte-RodríguezL. C. F. Ferreira and E. J. Villamizar-Roa, Global existence for an attraction-repulsion chemotaxis fluid model with logistic source, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 423-447.  doi: 10.3934/dcdsb.2018180.

[7]

H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.  doi: 10.1002/mana.19981950106.

[8]

Y. Guo and Y. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208.  doi: 10.1080/03605302.2012.696296.

[9]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 633-683. 

[10]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.

[11]

S. Ishida, Global existence and boundedness for chemotaxis–Navier–Stokes systems with position-dependent sensitivity in 2D bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 3463-3482.  doi: 10.3934/dcds.2015.35.3463.

[12]

C. Jin, Global classical solution and stability to a coupled chemotaxis–fluid model with logistic source, Discrete Contin. Dyn. Syst., 38 (2018), 3547-3566.  doi: 10.3934/dcds.2018150.

[13]

H.-Y. 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.

[14]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[15]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.

[16]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural' ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, RI, 1968.

[17]

M. LiuM. Yu and H. Luo, Global weak solution to the chemotaxis–fluid system, J. Math. Res. Appl., 39 (2019), 181-195. 

[18]

N. Masmoudi and F. Rousset, Uniform regularity for the Navier–Stokes equation with Navier boundary condition, Arch. Ration. Mech. Anal., 203 (2012), 529-575.  doi: 10.1007/s00205-011-0456-5.

[19]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. 

[20]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller–Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. 

[21]

L. Sundbye, Global existence for the Dirichlet problem for the viscous shallow water equations, J. Math. Anal. Appl., 202 (1996), 236-258.  doi: 10.1006/jmaa.1996.0315.

[22]

Q. Tao and Z. Yao, Global existence and large time behavior for a two-dimensional chemotaxis–shallow water system, J. Differential Equations, 265 (2018), 3092-3129.  doi: 10.1016/j.jde.2018.05.002.

[23]

R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York, 1979.

[24]

T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blow-up in a finite and the inifinite time, Methods Appl. Anal, 8 (2001), 349-367.  doi: 10.4310/MAA.2001.v8.n2.a9.

[25]

W. Wang and Y. Wang, The $L^p$ decay estimates for the chemotaxis–shallow water system, J. Math. Anal. Appl., 474 (2019), 640-665.  doi: 10.1016/j.jmaa.2019.01.066.

[26]

Y. Wang and X. Cao, Global classical solutions of a 3D chemotaxis–Stokes system with rotation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3235-3254.  doi: 10.3934/dcdsb.2015.20.3235.

[27]

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.

[28]

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.

[29]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Math. Pures Appl. (9), 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.

[30]

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.

[31]

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.

[32]

M. Winkler, Global mass-preserving solutions in a two-dimensional chemotaxis–Stokes system with rotational flux components, J. Evol. Equ., 18 (2018), 1267-1289.  doi: 10.1007/s00028-018-0440-8.

[33]

M. Winkler, Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis(–stokes) systems?, Int. Math. Res. Not., (2019). doi: 10.1093/imrn/rnz056.

[34]

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.

show all references

References:
[1]

X. Cao, Global bounded solutions of the higher-dimensional Keller–Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891.

[2]

X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier–Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differential Equations, 55 (2016), 39pp. doi: 10.1007/s00526-016-1027-2.

[3]

J. CheL. ChenB. Duan and Z. Luo, On the existence of local strong solutions to chemotaxis–shallow water system with large data and vacuum, J. Differential Equations, 261 (2016), 6758-6789.  doi: 10.1016/j.jde.2016.09.005.

[4]

Y. Cho and H. Kim, On classical solutions of the compressible Navier–Stokes equations with nonnegative initial densities, Manuscripta Math., 120 (2006), 91-129.  doi: 10.1007/s00229-006-0637-y.

[5]

R. DuanX. Li and Z. Xiang, Global existence and large time behavior for a two-dimensional chemotaxis–Navier–Stokes system, J. Differential Equations, 263 (2017), 6284-6316.  doi: 10.1016/j.jde.2017.07.015.

[6]

A. Duarte-RodríguezL. C. F. Ferreira and E. J. Villamizar-Roa, Global existence for an attraction-repulsion chemotaxis fluid model with logistic source, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 423-447.  doi: 10.3934/dcdsb.2018180.

[7]

H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.  doi: 10.1002/mana.19981950106.

[8]

Y. Guo and Y. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208.  doi: 10.1080/03605302.2012.696296.

[9]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 633-683. 

[10]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.

[11]

S. Ishida, Global existence and boundedness for chemotaxis–Navier–Stokes systems with position-dependent sensitivity in 2D bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 3463-3482.  doi: 10.3934/dcds.2015.35.3463.

[12]

C. Jin, Global classical solution and stability to a coupled chemotaxis–fluid model with logistic source, Discrete Contin. Dyn. Syst., 38 (2018), 3547-3566.  doi: 10.3934/dcds.2018150.

[13]

H.-Y. 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.

[14]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[15]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.

[16]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural' ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, RI, 1968.

[17]

M. LiuM. Yu and H. Luo, Global weak solution to the chemotaxis–fluid system, J. Math. Res. Appl., 39 (2019), 181-195. 

[18]

N. Masmoudi and F. Rousset, Uniform regularity for the Navier–Stokes equation with Navier boundary condition, Arch. Ration. Mech. Anal., 203 (2012), 529-575.  doi: 10.1007/s00205-011-0456-5.

[19]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. 

[20]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller–Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. 

[21]

L. Sundbye, Global existence for the Dirichlet problem for the viscous shallow water equations, J. Math. Anal. Appl., 202 (1996), 236-258.  doi: 10.1006/jmaa.1996.0315.

[22]

Q. Tao and Z. Yao, Global existence and large time behavior for a two-dimensional chemotaxis–shallow water system, J. Differential Equations, 265 (2018), 3092-3129.  doi: 10.1016/j.jde.2018.05.002.

[23]

R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York, 1979.

[24]

T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blow-up in a finite and the inifinite time, Methods Appl. Anal, 8 (2001), 349-367.  doi: 10.4310/MAA.2001.v8.n2.a9.

[25]

W. Wang and Y. Wang, The $L^p$ decay estimates for the chemotaxis–shallow water system, J. Math. Anal. Appl., 474 (2019), 640-665.  doi: 10.1016/j.jmaa.2019.01.066.

[26]

Y. Wang and X. Cao, Global classical solutions of a 3D chemotaxis–Stokes system with rotation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3235-3254.  doi: 10.3934/dcdsb.2015.20.3235.

[27]

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.

[28]

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.

[29]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Math. Pures Appl. (9), 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.

[30]

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.

[31]

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.

[32]

M. Winkler, Global mass-preserving solutions in a two-dimensional chemotaxis–Stokes system with rotational flux components, J. Evol. Equ., 18 (2018), 1267-1289.  doi: 10.1007/s00028-018-0440-8.

[33]

M. Winkler, Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis(–stokes) systems?, Int. Math. Res. Not., (2019). doi: 10.1093/imrn/rnz056.

[34]

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.

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