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May  2017, 37(5): 2829-2859. doi: 10.3934/dcds.2017122

Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data

Xiaoping Zhai 1, and  Zhaoyang Yin 1,2,

1. 

Department of Mathematics, Sun Yat-sen University, Guangzhou, Guangdong 510275, China
2. 

Faculty of Information Technology, Macau University of Science and Technology, Macau

Received  January 2016 Revised  December 2016 Published  February 2017

In this paper, we mainly study the Cauchy problem of the Chemo-taxis-Navier-Stokes equations with initial data in critical Besov spaces. We first get the local wellposedness of the system in $\mathbb{R}^d \, (d≥2)$ by the Picard theorem, and then extend the local solutions to be global under the only smallness assumptions on $\|u_0^h\|_{\dot{B}_{p, 1}^{-1+\frac{d}{p}}}$, $\|n_0\|_{\dot{B}_{q, 1}^{-2+\frac{d}{q}}}$ and $\|c_0\|_{\dot{B}_{r, 1}^{\frac{d}{r}}}$. This obtained result implies the global wellposedness of the equations with large initial vertical velocity component. Moreover, by fully using the global wellposedness of the classical 2D Navier-Stokes equations and the weighted Chemin-Lerner space, we can also extend the obtained local solutions to be global in $\mathbb{R}^2$ provided the initial cell density $n_0$ and the initial chemical concentration $c_0$ are doubly exponential small compared with the initial velocity field $u_0$.

Keywords: The Chemotaxis-Navier-Stokes equations, global wellposedness, Besov spaces.
Mathematics Subject Classification: Primary: 35Q53; Secondary: 35A01, 76D05, 92C17.
Citation: Xiaoping Zhai, Zhaoyang Yin. Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2829-2859. doi: 10.3934/dcds.2017122
References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations Grundlehren Math. Wiss. , 343 Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

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R. Danchin, Local theory in critical spaces for compressible viscous and heat-conducting gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233.   Google Scholar

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R. J. Duan, A. Lorz and P. A. 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

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M. Di Francesco, A. Lorz and P. A. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., 28 (2010), 1437-1453.   Google Scholar

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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

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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

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

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M. Paicu and P. Zhang, Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262 (2012), 3556-3584.  doi: 10.1016/j.jfa.2012.01.022.  Google Scholar

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Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.  doi: 10.1016/j.anihpc.2012.07.002.  Google Scholar

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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

[21]

Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.  doi: 10.1137/100802943.  Google Scholar

[22]

I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near constant lines, Proc. Natl. Acad. Sci., 102 (2005), 2277-2282.   Google Scholar

[23]

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

[24]

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

[25]

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

[26]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

[27]

C. Zhai and T. Zhang, Global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity, J. Math. Phys. , 56 (2015), 091512. doi: 10.1063/1.4931467.  Google Scholar

show all references

References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations Grundlehren Math. Wiss. , 343 Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., 14 (1981), 209-246.   Google Scholar

[3]

P. Biler and G. Karch, Blow-up of solutions to generalized Keller-Segel model, J. Evol. Equ., 10 (2010), 247-262.  doi: 10.1007/s00028-009-0048-0.  Google Scholar

[4]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 44 (2006), 1-32.   Google Scholar

[5]

V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $\mathbb{R}^2$, Commun. Math. Sci., 6 (2008), 417-447.  doi: 10.4310/CMS.2008.v6.n2.a8.  Google Scholar

[6]

M. Chae, K. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297.  doi: 10.3934/dcds.2013.33.2271.  Google Scholar

[7]

R. Danchin, Local theory in critical spaces for compressible viscous and heat-conducting gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233.   Google Scholar

[8]

R. J. Duan, A. Lorz and P. A. 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

[9]

M. Di Francesco, A. Lorz and P. A. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., 28 (2010), 1437-1453.   Google Scholar

[10]

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

[11]

J. Huang, M. Paicu and P. Zhang, Global solutions to 2-D inhomogeneous Navier-Stokes system with general velocity, J. Math. Pures Appl., 100 (2013), 806-831.  doi: 10.1016/j.matpur.2013.03.003.  Google Scholar

[12]

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.  Google Scholar

[13]

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

[14]

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

[15]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics 27, Cambridge University Press, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511613203.  Google Scholar

[16]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces, Comm. Math. Phys., 307 (2011), 713-759.  doi: 10.1007/s00220-011-1350-6.  Google Scholar

[17]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262 (2012), 3556-3584.  doi: 10.1016/j.jfa.2012.01.022.  Google Scholar

[18]

Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 1901-1914.   Google Scholar

[19]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.  doi: 10.1016/j.anihpc.2012.07.002.  Google Scholar

[20]

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

[21]

Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.  doi: 10.1137/100802943.  Google Scholar

[22]

I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near constant lines, Proc. Natl. Acad. Sci., 102 (2005), 2277-2282.   Google Scholar

[23]

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

[24]

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

[25]

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

[26]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

[27]

C. Zhai and T. Zhang, Global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity, J. Math. Phys. , 56 (2015), 091512. doi: 10.1063/1.4931467.  Google Scholar

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