October  2019, 24(10): 5409-5436. doi: 10.3934/dcdsb.2019064

Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux

1. 

School of Mathematics, Southeast University, Nanjing 210096, China

2. 

Institute for Applied Mathematics, School of Mathematics, Southeast University, Nanjing 211189, China

* Corresponding author

Received  June 2018 Revised  November 2018 Published  April 2019

Fund Project: The authors are supported in part by NSF of China (No. 11671079, No. 11701290, No. 11601127, No. 11871147 and No. 11871148), and NSF of Jiangsu Province (No. BK20170896).

In this paper, we consider the chemotaxis-Navier-Stokes system with nonlinear diffusion and rotational flux given by
$\begin{eqnarray*} \left\{\begin{array}{lll}n_{t}+u\cdot\nabla n=\Delta n^m-\nabla\cdot(uS(x,n,c)\cdot\nabla c),&x\in\Omega,\ \ t>0,\\[1mm]c_t+u\cdot\nabla c=\Delta c-c+n,&x\in\Omega,\ \ t>0,\\[1mm]u_t+k(u\cdot\nabla)u=\Delta u+\nabla P+n\nabla\phi,&x\in\Omega,\ \ t>0\\[1mm]\nabla\cdot u=0,&x\in\Omega,\ \ t>0 \end{array}\right.\end{eqnarray*}$
in a bounded domain
$\Omega\subset\mathbb{R}^3$
, where
$k\in\mathbb{R}$
,
$\phi\in W^{2,\infty}(\Omega)$
and the given tensor-valued function
$S$
:
$\overline\Omega\times[0,\infty)^2\rightarrow\mathbb{R}^{3\times 3}$
satisfies
$|S(x,n,c)|\leq S_0(n+1)^{-\alpha}\ \ {\rm for\ all}\ x\in\mathbb{R}^3,\ n\geq0,\ c\geq0.$
Imposing no restriction on the size of the initial data, we establish the global existence of a very weak solution while assuming
$m+\alpha>\frac{4}{3}$
and
$m>\frac{1}{3}$
.
Citation: 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
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J. Liu and Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 262 (2017), 5271-5305.  doi: 10.1016/j.jde.2017.01.024.  Google Scholar

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[26]

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[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.  Google Scholar

[28]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146.  Google Scholar

[29]

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

[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.  Google Scholar

[31]

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

[32]

M. Winkler, Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3092-3115.  doi: 10.1137/140979708.  Google Scholar

[33]

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

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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

[35]

M. Winkler, Does fluid interaction affect regularity in the three-dimensional Keller-Segel system with saturated sensitivity?, J. Math. Fluid. Mech., 20 (2018), 1889-1909.  doi: 10.1007/s00021-018-0395-0.  Google Scholar

[36]

M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differential Equations, 264 (2018), 6109-6151.  doi: 10.1016/j.jde.2018.01.027.  Google Scholar

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D. Wrzosek, Long-time behaviour of solutions to a chemotaxis model with volume-filling effect, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 431-444.  doi: 10.1017/S0308210500004649.  Google Scholar

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M. Yang, Global solutions to Keller-Segel-Navier-Stokes equations with a class of large initial data in critical Besov spaces, Math. Methods Appl. Sci., 40 (2017), 7425-7437.  doi: 10.1002/mma.4538.  Google Scholar

[39]

H. YuW. Wang and S. Zheng, Global classical solutions to the Keller-Segel-Navier-Stokes system with matrix-valued sensitivity, J. Math. Anal. Appl., 461 (2018), 1748-1770.  doi: 10.1016/j.jmaa.2017.12.048.  Google Scholar

[40]

Q. Zhang and Y. Li, Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl., 418 (2014), 47-63.  doi: 10.1016/j.jmaa.2014.03.084.  Google Scholar

[41]

Q. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.  doi: 10.1007/s00033-015-0532-z.  Google Scholar

[42]

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), 2527-2551.  doi: 10.3934/dcdsb.2015.20.2751.  Google Scholar

[43]

Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.  doi: 10.1007/s00033-013-0383-4.  Google Scholar

[44]

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

[45]

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

[46]

J. Zheng, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 263 (2017), 2606-2629.  doi: 10.1016/j.jde.2017.04.005.  Google Scholar

show all references

References:
[1]

V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl. (9), 86 (2006), 155-175.  doi: 10.1016/j.matpur.2006.04.002.  Google Scholar

[2]

X. Cao and S. Ishida, Global-in-time bounded weak solutions to a degenerate quasilinear Keller-Segel system with rotation, Nonlinearity, 27 (2014), 1899-1913.  doi: 10.1088/0951-7715/27/8/1899.  Google Scholar

[3]

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.  Google Scholar

[4]

R. DuanA. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673.  doi: 10.1080/03605302.2010.497199.  Google Scholar

[5]

H. He and Q. Zhang, Global existence of weak solutions for the 3D chemotaxis-Navier-Stokes equations, Nonlinear Anal. Real World Appl., 35 (2017), 336-349.  doi: 10.1016/j.nonrwa.2016.11.006.  Google Scholar

[6]

T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 280-301.  doi: 10.1006/aama.2001.0721.  Google Scholar

[7]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[8]

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

[9]

A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and enhancement of biological reactions, Comm. Partial Differential Equations, 37 (2012), 298-318.  doi: 10.1080/03605302.2011.589879.  Google Scholar

[10]

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

[11]

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

[12]

Y. Li and Y. Li, Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species, Nonlinear Anal., 109 (2014), 72-84.  doi: 10.1016/j.na.2014.05.021.  Google Scholar

[13]

Y. Li and Y. Li, Global boundedness of solutions for the chemotaxis-Navier-Stokes system in $\Bbb{R}^2$, J. Differential Equations, 261 (2016), 6570-6613.  doi: 10.1016/j.jde.2016.08.045.  Google Scholar

[14]

J. Liu and Y. Wang, Global existence and boundedness in a Keller-Segel-(Navier-)Stokes system with signal-dependent sensitivity, J. Math. Anal. Appl., 447 (2017), 499-528.  doi: 10.1016/j.jmaa.2016.10.028.  Google Scholar

[15]

J. Liu and Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 262 (2017), 5271-5305.  doi: 10.1016/j.jde.2017.01.024.  Google Scholar

[16]

J.-G. 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]

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

[18]

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

[19]

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, 26pp. doi: 10.1007/s00033-017-0816-6.  Google Scholar

[20]

Z. Tan and X. Zhang, Decay estimates of the coupled chemotaxis-fluid equations in R3, J. Math. Anal. Appl., 410 (2014), 27-38.  doi: 10.1016/j.jmaa.2013.08.008.  Google Scholar

[21]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[22]

Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.  Google Scholar

[23]

I. TuvalL. CisnerosC. DombrowskiC. W. WolgemuthJ. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc Natl Acad Sci U S A, 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102.  Google Scholar

[24]

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

[25]

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.  Google Scholar

[26]

Y. WangM. 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.   Google Scholar

[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.  Google Scholar

[28]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146.  Google Scholar

[29]

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

[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.  Google Scholar

[31]

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

[32]

M. Winkler, Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3092-3115.  doi: 10.1137/140979708.  Google Scholar

[33]

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

[34]

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

[35]

M. Winkler, Does fluid interaction affect regularity in the three-dimensional Keller-Segel system with saturated sensitivity?, J. Math. Fluid. Mech., 20 (2018), 1889-1909.  doi: 10.1007/s00021-018-0395-0.  Google Scholar

[36]

M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differential Equations, 264 (2018), 6109-6151.  doi: 10.1016/j.jde.2018.01.027.  Google Scholar

[37]

D. Wrzosek, Long-time behaviour of solutions to a chemotaxis model with volume-filling effect, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 431-444.  doi: 10.1017/S0308210500004649.  Google Scholar

[38]

M. Yang, Global solutions to Keller-Segel-Navier-Stokes equations with a class of large initial data in critical Besov spaces, Math. Methods Appl. Sci., 40 (2017), 7425-7437.  doi: 10.1002/mma.4538.  Google Scholar

[39]

H. YuW. Wang and S. Zheng, Global classical solutions to the Keller-Segel-Navier-Stokes system with matrix-valued sensitivity, J. Math. Anal. Appl., 461 (2018), 1748-1770.  doi: 10.1016/j.jmaa.2017.12.048.  Google Scholar

[40]

Q. Zhang and Y. Li, Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl., 418 (2014), 47-63.  doi: 10.1016/j.jmaa.2014.03.084.  Google Scholar

[41]

Q. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.  doi: 10.1007/s00033-015-0532-z.  Google Scholar

[42]

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), 2527-2551.  doi: 10.3934/dcdsb.2015.20.2751.  Google Scholar

[43]

Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.  doi: 10.1007/s00033-013-0383-4.  Google Scholar

[44]

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

[45]

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

[46]

J. Zheng, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 263 (2017), 2606-2629.  doi: 10.1016/j.jde.2017.04.005.  Google Scholar

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