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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

Myeongju Chae 1, , Kyungkeun Kang 2, and  Jihoon Lee 3,

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.
Keywords: energy estimates., Chemotaxis-fluid equations, Keller-Segel, global solutions, Navier-Stokes system.
Mathematics Subject Classification: Primary: 35Q30, 35Q35; Secondary: 35Q92, 76B0.
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.

show all references

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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