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

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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

Yulan Wang ^1, and Xinru Cao ^2,

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

Abstract
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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 $\frac{1}{6}$.

Keywords: boundedness, Chemotaxis, Stokes, global existence, rotation..

Mathematics Subject Classification: Primary: 35K55, 35Q92; Secondary: 35Q35, 92C1.

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.
[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.

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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