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November  2015, 20(9): 3235-3254. doi: 10.3934/dcdsb.2015.20.3235

Global classical solutions of a 3D chemotaxis-Stokes system with rotation

1. 

School of Science, Xihua University, Chengdu 610039, China

2. 

Institut für Mathematik, Universität Paderborn, Paderborn 33098, Germany

Received  September 2014 Revised  December 2014 Published  September 2015

This paper considers the chemotaxis-Stokes system $$\begin{cases} \displaystyle n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\cdot\nabla c), &(x,t)\in \Omega\times (0,T),\\ \displaystyle c_t+u\cdot\nabla c=\Delta c-nc, &(x,t)\in\Omega\times (0,T),\qquad(\star)\\ \displaystyle u_t=\Delta u+\nabla P+n\nabla\phi , &(x,t)\in\Omega\times (0,T),\\ \nabla\cdot u=0,&(x,t)\in\Omega\times (0,T). \end{cases}$$ under no-flux boundary conditions in a bounded domain $\Omega \subset \mathbb{R}^3$ with smooth boundary. Here $S$ is a matrix-valued sensitivity satisfying $|S(x,n,c)|<\tilde{C}(1+n)^{-\alpha}$ with some $\tilde{C}>0$ and $\alpha>0$. Although $(\star)$ does not possess the natural gradient-like functional structure available when $S$ reduces to a scalar function, we can still establish a new energy type inequality. Based on this inequality we achieve a coupled estimate for arbitrarily high Lebesgue norms of $n$ and $\nabla c$. This helps us to finally obtain the existence of a global classical solution when $\alpha$ is bigger than $\frac{1}{6}$.
Citation: Yulan Wang, Xinru Cao. Global classical solutions of a 3D chemotaxis-Stokes system with rotation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3235-3254. doi: 10.3934/dcdsb.2015.20.3235
References:
[1]

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.

[2]

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

[3]

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.

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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.

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Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3.

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Y. Giga and H. Sohr, Abstract $L^p$ estimate for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94. doi: 10.1016/0022-1236(91)90136-S.

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin/Heidelberg, 1981.

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S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010. doi: 10.1016/j.jde.2014.01.028.

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T. Li, A. Suen, M. Winkler and C. Xue, Gobal small-data solutions in a chemotaxis system with rotation, Math. Mod. Meth. Appl. Sci., 25(2015), 721-747.

[10]

J. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence, Ann. I. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652. doi: 10.1016/j.anihpc.2011.04.005.

[11]

J. L. Lions, Équations Différentielles Opérationnelles et Problémes aux Limites, Die Grundlehren der mathematischen Wissenschaften, Springer, 1961.

[12]

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

[13]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasi-Linear Equations of Parabolic Type, Amer. Math. Soc. Transl., Providence, RI, 1968.

[14]

Y. Lou, Y. Tao and M. Winkler, Approching the ideal free distribution in two-species copetition models with fitness-dependent dispersal, SIAM J. Math. Anal., 46 (2014), 1228-1262. doi: 10.1137/130934246.

[15]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.

[16]

M. M. Porzio and V. Vespri, Hölder estimate for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045.

[17]

P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up,Global Existence and Steady States, Birkhäuser Advanced Texts, Basel/Boston/Berlin, 2007.

[18]

H. Sohr, The Navier-Stokes Equations. an Elementary Functional Analytic Approach, Birkhăuser, Basel, 2001. doi: 10.1007/978-3-0348-8255-2.

[19]

Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529. doi: 10.1016/j.jmaa.2011.02.041.

[20]

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.

[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.

[22]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543. doi: 10.1016/j.jde.2011.07.010.

[23]

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

[24]

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

[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.

[26]

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.

[27]

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.

[28]

M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, (2015), arXiv:1410.5929. doi: 10.1016/j.anihpc.2015.05.002.

[29]

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.

[30]

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]

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.

[2]

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

[3]

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.

[4]

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.

[5]

Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3.

[6]

Y. Giga and H. Sohr, Abstract $L^p$ estimate for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94. doi: 10.1016/0022-1236(91)90136-S.

[7]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin/Heidelberg, 1981.

[8]

S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010. doi: 10.1016/j.jde.2014.01.028.

[9]

T. Li, A. Suen, M. Winkler and C. Xue, Gobal small-data solutions in a chemotaxis system with rotation, Math. Mod. Meth. Appl. Sci., 25(2015), 721-747.

[10]

J. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence, Ann. I. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652. doi: 10.1016/j.anihpc.2011.04.005.

[11]

J. L. Lions, Équations Différentielles Opérationnelles et Problémes aux Limites, Die Grundlehren der mathematischen Wissenschaften, Springer, 1961.

[12]

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

[13]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasi-Linear Equations of Parabolic Type, Amer. Math. Soc. Transl., Providence, RI, 1968.

[14]

Y. Lou, Y. Tao and M. Winkler, Approching the ideal free distribution in two-species copetition models with fitness-dependent dispersal, SIAM J. Math. Anal., 46 (2014), 1228-1262. doi: 10.1137/130934246.

[15]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.

[16]

M. M. Porzio and V. Vespri, Hölder estimate for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045.

[17]

P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up,Global Existence and Steady States, Birkhäuser Advanced Texts, Basel/Boston/Berlin, 2007.

[18]

H. Sohr, The Navier-Stokes Equations. an Elementary Functional Analytic Approach, Birkhăuser, Basel, 2001. doi: 10.1007/978-3-0348-8255-2.

[19]

Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529. doi: 10.1016/j.jmaa.2011.02.041.

[20]

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.

[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.

[22]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543. doi: 10.1016/j.jde.2011.07.010.

[23]

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

[24]

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

[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.

[26]

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.

[27]

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.

[28]

M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, (2015), arXiv:1410.5929. doi: 10.1016/j.anihpc.2015.05.002.

[29]

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.

[30]

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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