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

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doi: 10.3934/dcdss.2020019

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

Yulan Wang ^,

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.

Keywords: Chemotaxis-Navier-Stokes system, self-consistent, nonlinear diffusion, global existence.

Mathematics Subject Classification: 35K55, 35Q92, 35Q35, 92C17.

Citation: Yulan Wang. Global solvability in a two-dimensional self-consistent chemotaxis-Navier-Stokes system. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020019

References:

[1]	N. Bellomo, A. Bellouquid, Y. 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. Google Scholar
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[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. Google Scholar
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[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. Google Scholar
[14]	X. Li, Y. 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. Google Scholar
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[17]	A. Lorz, Coupled chemotaxis fluid equations, Math. Models Methods Appl. Sci., 20 (2010), 987-1004. doi: 10.1142/S0218202510004507. Google Scholar
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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. Google Scholar
[20]	Y. Shibata and S. Shimizu, On the L^p − L^q 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. Google Scholar
[21]	H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach, Birkh$\check{a}$user, Basel, 2001. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[31]	Y. Wang, M. 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. Google Scholar
[32]	Y. Wang, M. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

show all references

References:

[1]	N. Bellomo, A. Bellouquid, Y. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	M. Chae, K. 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. Google Scholar
[6]	M. Chae, K. 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. Google Scholar
[7]	M. Difrancesco, A. 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. Google Scholar
[8]	R. Duan, X. 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. Google Scholar
[9]	R. Duan, A. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	X. Li, Y. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[17]	A. Lorz, Coupled chemotaxis fluid equations, Math. Models Methods Appl. Sci., 20 (2010), 987-1004. doi: 10.1142/S0218202510004507. Google Scholar
[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. Google Scholar
[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. Google Scholar
[20]	Y. Shibata and S. Shimizu, On the L^p − L^q 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. Google Scholar
[21]	H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach, Birkh$\check{a}$user, Basel, 2001. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[31]	Y. Wang, M. 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. Google Scholar
[32]	Y. Wang, M. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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[19]	Lukáš Poul. Existence of weak solutions to the Navier-Stokes-Fourier system on Lipschitz domains. Conference Publications, 2007, 2007 (Special) : 834-843. doi: 10.3934/proc.2007.2007.834
[20]	Hammadi Abidi, Taoufik Hmidi, Sahbi Keraani. On the global regularity of axisymmetric Navier-Stokes-Boussinesq system. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 737-756. doi: 10.3934/dcds.2011.29.737

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