Existence of smooth solutions to coupled chemotaxis-fluid equations

Previous Article
Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle
DCDS Home
This Issue
Next Article
Structure of solutions to a singular Liouville system arising from modeling dissipative stationary plasmas

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

Abstract
Related Papers

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 & Continuous Dynamical Systems - A, 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, (2006). Google Scholar
[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. Google Scholar
[3]	L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis,, C. R. Math. Acad. Sci. Paris, 336 (2003), 141. doi: 10.1016/S1631-073X(02)00008-0. Google Scholar
[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. doi: 10.1007/s00032-003-0026-x. Google Scholar
[5]	R. DiPerna and P. L. Lions, On the Cauchy problem for Blotzmann equations: Global existence and weak stability,, Ann. Math., 139 (1989), 321. doi: 10.2307/1971423. Google Scholar
[6]	R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations,, Comm. Partial Diff. Equations, 35 (2010), 1635. doi: 10.1080/03605302.2010.497199. Google Scholar
[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. doi: 10.3934/dcds.2010.28.1437. Google Scholar
[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. doi: 10.1016/0022-1236(91)90136-S. Google Scholar
[9]	M. A. Herrero and J. L. L. Velazquez, A blow-up mechanism for chemotaxis model,, Ann. Sc. Norm. Super. Pisa, 24 (1997), 633. Google Scholar
[10]	D. Horstman and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions,, European J. Appl. Math, 12 (2001), 159. doi: 10.1017/S0956792501004363. Google Scholar
[11]	E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewd as an instability,, J. Theor. Biol., 26 (1970), 399. Google Scholar
[12]	E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225. Google Scholar
[13]	J.-G. Liu and A. Lorz, A coupled chemotaxis-fluid model,, I. H. Poincaré, 28 (2011), 643. doi: 10.1016/j.anihpc.2011.04.005. Google Scholar
[14]	A. Lorz, Coupled chemotaxis fluid model,, Math. Models and Meth. in Appl. Sci., 20 (2010), 987. doi: 10.1142/S0218202510004507. Google Scholar
[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. Google Scholar
[16]	K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations,, Funkcial Ekvac., 44 (2001), 441. Google Scholar
[17]	C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biol. Biophys., 15 (1953), 311. Google Scholar
[18]	L. Tartar, "Topics in Nonlinear Analysis,", Publicatons mathématiques de l'Université de Paris-Sud(Orsay), (1978). Google Scholar
[19]	Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria,, J. Math. Anal. Appl., 381 (2011), 521. doi: 10.1016/j.jmaa.2011.02.041. Google Scholar
[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. doi: 10.3934/dcds.2012.32.1901. Google Scholar
[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. Google Scholar
[22]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008. Google Scholar
[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. doi: 10.1080/03605302.2011.591865. Google Scholar

show all references

References:

[1]	J. Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, "Mathematical Geophysics,", Oxford university press, (2006). Google Scholar
[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. Google Scholar
[3]	L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis,, C. R. Math. Acad. Sci. Paris, 336 (2003), 141. doi: 10.1016/S1631-073X(02)00008-0. Google Scholar
[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. doi: 10.1007/s00032-003-0026-x. Google Scholar
[5]	R. DiPerna and P. L. Lions, On the Cauchy problem for Blotzmann equations: Global existence and weak stability,, Ann. Math., 139 (1989), 321. doi: 10.2307/1971423. Google Scholar
[6]	R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations,, Comm. Partial Diff. Equations, 35 (2010), 1635. doi: 10.1080/03605302.2010.497199. Google Scholar
[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. doi: 10.3934/dcds.2010.28.1437. Google Scholar
[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. doi: 10.1016/0022-1236(91)90136-S. Google Scholar
[9]	M. A. Herrero and J. L. L. Velazquez, A blow-up mechanism for chemotaxis model,, Ann. Sc. Norm. Super. Pisa, 24 (1997), 633. Google Scholar
[10]	D. Horstman and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions,, European J. Appl. Math, 12 (2001), 159. doi: 10.1017/S0956792501004363. Google Scholar
[11]	E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewd as an instability,, J. Theor. Biol., 26 (1970), 399. Google Scholar
[12]	E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225. Google Scholar
[13]	J.-G. Liu and A. Lorz, A coupled chemotaxis-fluid model,, I. H. Poincaré, 28 (2011), 643. doi: 10.1016/j.anihpc.2011.04.005. Google Scholar
[14]	A. Lorz, Coupled chemotaxis fluid model,, Math. Models and Meth. in Appl. Sci., 20 (2010), 987. doi: 10.1142/S0218202510004507. Google Scholar
[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. Google Scholar
[16]	K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations,, Funkcial Ekvac., 44 (2001), 441. Google Scholar
[17]	C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biol. Biophys., 15 (1953), 311. Google Scholar
[18]	L. Tartar, "Topics in Nonlinear Analysis,", Publicatons mathématiques de l'Université de Paris-Sud(Orsay), (1978). Google Scholar
[19]	Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria,, J. Math. Anal. Appl., 381 (2011), 521. doi: 10.1016/j.jmaa.2011.02.041. Google Scholar
[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. doi: 10.3934/dcds.2012.32.1901. Google Scholar
[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. Google Scholar
[22]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008. Google Scholar
[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. doi: 10.1080/03605302.2011.591865. Google Scholar

[1]	Hi Jun Choe, Bataa Lkhagvasuren, Minsuk Yang. Wellposedness of the Keller-Segel Navier-Stokes equations in the critical Besov spaces. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2453-2464. doi: 10.3934/cpaa.2015.14.2453
[2]	Hao Yu, Wei Wang, Sining Zheng. Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1317-1327. doi: 10.3934/dcdsb.2016.21.1317
[3]	Qi Wang. Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1231-1250. doi: 10.3934/dcdsb.2015.20.1231
[4]	Xinru Cao. Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1891-1904. doi: 10.3934/dcds.2015.35.1891
[5]	Piotr Biler, Ignacio Guerra, Grzegorz Karch. Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2117-2126. doi: 10.3934/cpaa.2015.14.2117
[6]	Tobias Black. Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 119-137. doi: 10.3934/dcdss.2020007
[7]	Hao Yu, Wei Wang, Sining Zheng. Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1635-1644. doi: 10.3934/dcdsb.2017078
[8]	Luca Battaglia. A general existence result for stationary solutions to the Keller-Segel system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 905-926. doi: 10.3934/dcds.2019038
[9]	Peter Anthony, Sergey Zelik. Infinite-energy solutions for the Navier-Stokes equations in a strip revisited. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1361-1393. doi: 10.3934/cpaa.2014.13.1361
[10]	Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure & Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243
[11]	Minghua Yang, Zunwei Fu, Jinyi Sun. Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3427-3460. doi: 10.3934/dcdsb.2018284
[12]	Norikazu Saito. Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis. Communications on Pure & Applied Analysis, 2012, 11 (1) : 339-364. doi: 10.3934/cpaa.2012.11.339
[13]	Zhichun Zhai. Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces. Communications on Pure & Applied Analysis, 2011, 10 (1) : 287-308. doi: 10.3934/cpaa.2011.10.287
[14]	Joanna Rencławowicz, Wojciech M. Zajączkowski. Global regular solutions to the Navier-Stokes equations with large flux. Conference Publications, 2011, 2011 (Special) : 1234-1243. doi: 10.3934/proc.2011.2011.1234
[15]	Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085
[16]	Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations & Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217
[17]	Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-28. doi: 10.3934/dcdsb.2019064
[18]	Luis Almeida, Federica Bubba, Benoît Perthame, Camille Pouchol. Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations. Networks & Heterogeneous Media, 2019, 14 (1) : 23-41. doi: 10.3934/nhm.2019002
[19]	Yadong Shang, Jianjun Paul Tian, Bixiang Wang. Asymptotic behavior of the stochastic Keller-Segel equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1367-1391. doi: 10.3934/dcdsb.2019020
[20]	Hui Huang, Jian-Guo Liu. Error estimates of the aggregation-diffusion splitting algorithms for the Keller-Segel equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3463-3478. doi: 10.3934/dcdsb.2016107

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences