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June  2013, 33(6): 2271-2297. doi: 10.3934/dcds.2013.33.2271

## Existence of smooth solutions to coupled chemotaxis-fluid equations

 1 Department of Applied Mathematics, Hankyong National University, Ansung, South Korea 2 Department of Mathematics, Yonsei University, Seoul, South Korea 3 Department of Mathematics, Sungkyunkwan University, Suwon, South Korea

Received  February 2012 Revised  July 2012 Published  December 2012

We consider a system coupling the parabolic-parabolic chemotaxis equations to the incompressible Navier-Stokes equations in spatial dimensions two and three. We establish the local existence of regular solutions and present some blow-up criterions. For two dimensional chemotaxis-Navier-Stokes equations, regular solutions constructed locally in time are, in reality, extended globally under some assumptions pertinent to experimental observations in [21] on the consumption rate and chemotactic sensitivity. We also show the existence of global weak solutions in spatially three dimensions with stronger restriction on the consumption rate and chemotactic sensitivity.
Citation: Myeongju Chae, Kyungkeun Kang, Jihoon Lee. Existence of smooth solutions to coupled chemotaxis-fluid equations. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2271-2297. doi: 10.3934/dcds.2013.33.2271
##### References:
 [1] J. Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, "Mathematical Geophysics," Oxford university press, London, 2006. [2] A. Chertock, K. Fellner, A. Kurganov, A. Lorz and P. A. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: A high-resolution numerical approach, to appear in J. Fluid Mech. doi: 10.1017/jfm.2011.534. [3] L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Math. Acad. Sci. Paris, 336 (2003), 141-146. doi: 10.1016/S1631-073X(02)00008-0. [4] L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x. [5] R. DiPerna and P. L. Lions, On the Cauchy problem for Blotzmann equations: Global existence and weak stability, Ann. Math., 139 (1989), 321-366. doi: 10.2307/1971423. [6] R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Diff. Equations, 35 (2010), 1635-1673. doi: 10.1080/03605302.2010.497199. [7] M. D. Francesco, A. Lorz and P. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Continuous Dynam. Systems - A, 28 (2010), 1437-1453. doi: 10.3934/dcds.2010.28.1437. [8] Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications fo Navier-Stokes equations in exterior domains, J. Funct. Analysis, 102 (1991), 72-94. doi: 10.1016/0022-1236(91)90136-S. [9] M. A. Herrero and J. L. L. Velazquez, A blow-up mechanism for chemotaxis model, Ann. Sc. Norm. Super. Pisa, 24 (1997), 633-683. [10] D. Horstman and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math, 12 (2001), 159-177. doi: 10.1017/S0956792501004363. [11] E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewd as an instability, J. Theor. Biol., 26 (1970), 399-415. [12] E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. [13] J.-G. Liu and A. Lorz, A coupled chemotaxis-fluid model, I. H. Poincaré, Analyse Non Linéaire, 28 (2011), 643-652. doi: 10.1016/j.anihpc.2011.04.005. [14] A. Lorz, Coupled chemotaxis fluid model, Math. Models and Meth. in Appl. Sci., 20 (2010), 987-1004. doi: 10.1142/S0218202510004507. [15] T. Nagai, T. Senba and K. Yoshida, Applications of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial Ekvac., 40 (1997), 411-433. [16] K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial Ekvac., 44 (2001), 441-469. [17] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biol. Biophys., 15 (1953), 311-338. [18] L. Tartar, "Topics in Nonlinear Analysis," Publicatons mathématiques de l'Université de Paris-Sud(Orsay), Paris, 1978. [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, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Continuous Dynam. Systems - A, 32 (2012), 1901-1914. doi: 10.3934/dcds.2012.32.1901. [21] I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, PNAS, 102 (2005), 2277-2282. [22] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2995. doi: 10.1016/j.jde.2010.02.008. [23] M. Winkler, Global large data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Diff. Equations, 37 (2012), 319-351. doi: 10.1080/03605302.2011.591865.

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##### References:
 [1] J. Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, "Mathematical Geophysics," Oxford university press, London, 2006. [2] A. Chertock, K. Fellner, A. Kurganov, A. Lorz and P. A. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: A high-resolution numerical approach, to appear in J. Fluid Mech. doi: 10.1017/jfm.2011.534. [3] L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Math. Acad. Sci. Paris, 336 (2003), 141-146. doi: 10.1016/S1631-073X(02)00008-0. [4] L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x. [5] R. DiPerna and P. L. Lions, On the Cauchy problem for Blotzmann equations: Global existence and weak stability, Ann. Math., 139 (1989), 321-366. doi: 10.2307/1971423. [6] R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Diff. Equations, 35 (2010), 1635-1673. doi: 10.1080/03605302.2010.497199. [7] M. D. Francesco, A. Lorz and P. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Continuous Dynam. Systems - A, 28 (2010), 1437-1453. doi: 10.3934/dcds.2010.28.1437. [8] Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications fo Navier-Stokes equations in exterior domains, J. Funct. Analysis, 102 (1991), 72-94. doi: 10.1016/0022-1236(91)90136-S. [9] M. A. Herrero and J. L. L. Velazquez, A blow-up mechanism for chemotaxis model, Ann. Sc. Norm. Super. Pisa, 24 (1997), 633-683. [10] D. Horstman and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math, 12 (2001), 159-177. doi: 10.1017/S0956792501004363. [11] E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewd as an instability, J. Theor. Biol., 26 (1970), 399-415. [12] E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. [13] J.-G. Liu and A. Lorz, A coupled chemotaxis-fluid model, I. H. Poincaré, Analyse Non Linéaire, 28 (2011), 643-652. doi: 10.1016/j.anihpc.2011.04.005. [14] A. Lorz, Coupled chemotaxis fluid model, Math. Models and Meth. in Appl. Sci., 20 (2010), 987-1004. doi: 10.1142/S0218202510004507. [15] T. Nagai, T. Senba and K. Yoshida, Applications of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial Ekvac., 40 (1997), 411-433. [16] K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial Ekvac., 44 (2001), 441-469. [17] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biol. Biophys., 15 (1953), 311-338. [18] L. Tartar, "Topics in Nonlinear Analysis," Publicatons mathématiques de l'Université de Paris-Sud(Orsay), Paris, 1978. [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, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Continuous Dynam. Systems - A, 32 (2012), 1901-1914. doi: 10.3934/dcds.2012.32.1901. [21] I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, PNAS, 102 (2005), 2277-2282. [22] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2995. doi: 10.1016/j.jde.2010.02.008. [23] M. Winkler, Global large data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Diff. Equations, 37 (2012), 319-351. doi: 10.1080/03605302.2011.591865.
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