February  2020, 13(2): 329-349. doi: 10.3934/dcdss.2020019

Global solvability in a two-dimensional self-consistent chemotaxis-Navier-Stokes system

School of Science, Xihua University, Chengdu 610039, China

* Corresponding author: wangyulan-math@163.com

Received  May 2017 Revised  October 2017 Published  January 2019

In this paper we deal with the initial-boundary value problem for chemotaxis-fluid model involving more complicated nonlinear coupling term, precisely, the following self-consistent system
$\left\{ \begin{array}{l}{n_t} + u \cdot \nabla n = \Delta {n^m} - \nabla \cdot (n\nabla c) + \nabla \cdot (n\nabla \phi ), \;\;\;\;\;\;(x, t) \in \Omega \times (0, T), \\{c_t} + u \cdot \nabla c = \Delta c - nc, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(x, t) \in \Omega \times (0, T), \\{u_t} + (u \cdot \nabla )u + \nabla P = \Delta u - n\nabla \phi + n\nabla c, \;\;\;\;\;\;\;\;\;\;\;\;\;(x, t) \in \Omega \times (0, T), \\\nabla \cdot u = 0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(x, t) \in \Omega \times (0, T), \end{array} \right.$
where
$Ω\subset \mathbb{R}^2$
is a bounded domain with smooth boundary.
The novelty here is that both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid is considered, which leads to the stronger coupling than usual chemotaxis-fluid model studied in the most existing literatures. To the best of our knowledge, there is no global solvability result on this chemotaxis-Navier-Stokes system in the past works. It is proved in this paper that global weak solutions exist whenever
$m>1$
and the initial data is suitably regular. This extends a result by Di Francesco, Lorz and Markowich (Discrete Cont. Dyn. Syst. A 28 (2010)) which asserts global existence of weak solutions under the constraint
$m∈(\frac{3}{2}, 2]$
in the corresponding Stokes-type simplified system.
Citation: Yulan Wang. Global solvability in a two-dimensional self-consistent chemotaxis-Navier-Stokes system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 329-349. doi: 10.3934/dcdss.2020019
References:
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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), Art. 107, 39 pp. doi: 10.1007/s00526-016-1027-2.

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M. ChaeK. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Discr. Cont. Dyn. Syst. A, 33 (2013), 2271-2297.  doi: 10.3934/dcds.2013.33.2271.

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M. ChaeK. Kang and J. Lee, Global Existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Part. Diff. Eqs., 39 (2014), 1205-1235.  doi: 10.1080/03605302.2013.852224.

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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.

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R. DuanA. Lorz and P. A. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Part. Diff. Eqs., 35 (2010), 1635-1673.  doi: 10.1080/03605302.2010.497199.

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R. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Not. IMRN, 2014 (2014), 1833-1852.  doi: 10.1093/imrn/rns270.

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S. Ishida, Global existence and Boundedness for chemotaxis-Navier-Stokes system with position-dependent sensitivity in 2D bounded domains, Discr. Cont. Dyn. Syst. A, 35 (2015), 3463-3482.  doi: 10.3934/dcds.2015.35.3463.

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X. LiY. Wang and Z. Xiang, Global existence and boundedness in a 2D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux, Commun. Math. Sci., 14 (2016), 1889-1910.  doi: 10.4310/CMS.2016.v14.n7.a5.

[15]

X. Li and Y. Xiao, Global existence and boundedness in a 2D Keller-Segel-Stokes system, Nonlinear Analysis-RWA, 37 (2017), 14-30.  doi: 10.1016/j.nonrwa.2017.02.005.

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[19]

Y. Peng and Z. Xiang, Global solutions to the coupled chemotaxis-fluids system in a 3D unbounded domain with boundary, Math. Models Methods Appl. Sci., 28 (2018), 869-920.  doi: 10.1142/S0218202518500239.

[20]

Y. Shibata and S. Shimizu, On the LpLq maximal regularity of Neumann problem for the Stokes equations in a bounded domain, J.Reine Angew.Math., 615 (2008), 157-209.  doi: 10.1515/CRELLE.2008.013.

[21]

H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach, Birkh$\check{a}$user, Basel, 2001.

[22]

Z. Tan and X. Zhang, Decay estimates of the coupled chemotaxis-fluid equations in $R^3$, J. Math. Anal. Appl., 410 (2014), 27-38.  doi: 10.1016/j.jmaa.2013.08.008.

[23]

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

[24]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxisStokes system with nonlinear diffusion, Ann. I. H. Poincar$\acute{e}$-AN, 30 (2013), 157-178.  doi: 10.1016/j.anihpc.2012.07.002.

[25]

I. Tuval, L. Cisneros and C. Dombrowski, et al., Bacterial swimming and oxygen transport near contact lines, Proc. Nat. Acad. Sci. USA., 102 (2005), 2277–2282. doi: 10.1073/pnas.0406724102.

[26]

J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.

[27]

D. Vorotnikov, Weak solutions for a bioconvection model related to Bacillus subtilis, Commun. Math. Sci., 12 (2014), 545-563.  doi: 10.4310/CMS.2014.v12.n3.a8.

[28]

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

[29]

Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: The 3D case, J.Differential Equations, 261 (2016), 4944-4973.  doi: 10.1016/j.jde.2016.07.010.

[30]

Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with subcritical sensitivity, Math. Models Methods Appl. Sci., 27 (2017), 2745-2780.  doi: 10.1142/S0218202517500579.

[31]

Y. WangM. Winkler and Z. Xiang, Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 18 (2018), 421-466.  doi: 10.2422/2036-2145.201603_004.

[32]

Y. WangM. Winkler and Z. Xiang, The small-convection limit in a two-dimensional chemotaxis-Navier-Stokes system, Math. Zeit., 289 (2018), 71-108.  doi: 10.1007/s00209-017-1944-6.

[33]

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.

[34]

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

[35]

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.

[36]

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.

[37]

M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var.Partial Differential Equations, 54 (2015), 3789-3828.  doi: 10.1007/s00526-015-0922-2.

[38]

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

[39]

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.

[40]

C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial population, SIAM J. Appl. Math., 70 (2009), 133-167.  doi: 10.1137/070711505.

[41]

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

[42]

Q. Zhang and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 259 (2015), 3730-3754.  doi: 10.1016/j.jde.2015.05.012.

[43]

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]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of KellerSegel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[2]

V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math.Pures Appl., 86 (2006), 155-175.  doi: 10.1016/j.matpur.2006.04.002.

[3]

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.

[4]

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), Art. 107, 39 pp. doi: 10.1007/s00526-016-1027-2.

[5]

M. ChaeK. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Discr. Cont. Dyn. Syst. A, 33 (2013), 2271-2297.  doi: 10.3934/dcds.2013.33.2271.

[6]

M. ChaeK. Kang and J. Lee, Global Existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Part. Diff. Eqs., 39 (2014), 1205-1235.  doi: 10.1080/03605302.2013.852224.

[7]

M. DifrancescoA. Lorz and P. A. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior, Discr. Cont. Dyn. Syst. A, 28 (2010), 1437-1453.  doi: 10.3934/dcds.2010.28.1437.

[8]

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.

[9]

R. DuanA. Lorz and P. A. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Part. Diff. Eqs., 35 (2010), 1635-1673.  doi: 10.1080/03605302.2010.497199.

[10]

R. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Not. IMRN, 2014 (2014), 1833-1852.  doi: 10.1093/imrn/rns270.

[11]

S. Ishida, Global existence and Boundedness for chemotaxis-Navier-Stokes system with position-dependent sensitivity in 2D bounded domains, Discr. Cont. Dyn. Syst. A, 35 (2015), 3463-3482.  doi: 10.3934/dcds.2015.35.3463.

[12]

R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588.  doi: 10.1016/j.jmaa.2004.12.009.

[13]

J. Lankeit, Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26 (2016), 2071-2109.  doi: 10.1142/S021820251640008X.

[14]

X. LiY. Wang and Z. Xiang, Global existence and boundedness in a 2D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux, Commun. Math. Sci., 14 (2016), 1889-1910.  doi: 10.4310/CMS.2016.v14.n7.a5.

[15]

X. Li and Y. Xiao, Global existence and boundedness in a 2D Keller-Segel-Stokes system, Nonlinear Analysis-RWA, 37 (2017), 14-30.  doi: 10.1016/j.nonrwa.2017.02.005.

[16]

J. 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.

[17]

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

[18]

Y. Peng and Z. Xiang, Global existence and boundedness in a 3D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux, Z. Angew. Math. Phys., 68 (2017), Art. 68, 26 pp. doi: 10.1007/s00033-017-0816-6.

[19]

Y. Peng and Z. Xiang, Global solutions to the coupled chemotaxis-fluids system in a 3D unbounded domain with boundary, Math. Models Methods Appl. Sci., 28 (2018), 869-920.  doi: 10.1142/S0218202518500239.

[20]

Y. Shibata and S. Shimizu, On the LpLq maximal regularity of Neumann problem for the Stokes equations in a bounded domain, J.Reine Angew.Math., 615 (2008), 157-209.  doi: 10.1515/CRELLE.2008.013.

[21]

H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach, Birkh$\check{a}$user, Basel, 2001.

[22]

Z. Tan and X. Zhang, Decay estimates of the coupled chemotaxis-fluid equations in $R^3$, J. Math. Anal. Appl., 410 (2014), 27-38.  doi: 10.1016/j.jmaa.2013.08.008.

[23]

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

[24]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxisStokes system with nonlinear diffusion, Ann. I. H. Poincar$\acute{e}$-AN, 30 (2013), 157-178.  doi: 10.1016/j.anihpc.2012.07.002.

[25]

I. Tuval, L. Cisneros and C. Dombrowski, et al., Bacterial swimming and oxygen transport near contact lines, Proc. Nat. Acad. Sci. USA., 102 (2005), 2277–2282. doi: 10.1073/pnas.0406724102.

[26]

J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.

[27]

D. Vorotnikov, Weak solutions for a bioconvection model related to Bacillus subtilis, Commun. Math. Sci., 12 (2014), 545-563.  doi: 10.4310/CMS.2014.v12.n3.a8.

[28]

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

[29]

Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: The 3D case, J.Differential Equations, 261 (2016), 4944-4973.  doi: 10.1016/j.jde.2016.07.010.

[30]

Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with subcritical sensitivity, Math. Models Methods Appl. Sci., 27 (2017), 2745-2780.  doi: 10.1142/S0218202517500579.

[31]

Y. WangM. Winkler and Z. Xiang, Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 18 (2018), 421-466.  doi: 10.2422/2036-2145.201603_004.

[32]

Y. WangM. Winkler and Z. Xiang, The small-convection limit in a two-dimensional chemotaxis-Navier-Stokes system, Math. Zeit., 289 (2018), 71-108.  doi: 10.1007/s00209-017-1944-6.

[33]

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.

[34]

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

[35]

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.

[36]

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.

[37]

M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var.Partial Differential Equations, 54 (2015), 3789-3828.  doi: 10.1007/s00526-015-0922-2.

[38]

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

[39]

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.

[40]

C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial population, SIAM J. Appl. Math., 70 (2009), 133-167.  doi: 10.1137/070711505.

[41]

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

[42]

Q. Zhang and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 259 (2015), 3730-3754.  doi: 10.1016/j.jde.2015.05.012.

[43]

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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