-
Previous Article
Structure of solutions to a singular Liouville system arising from modeling dissipative stationary plasmas
- DCDS Home
- This Issue
-
Next Article
Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle
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 |
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. |
[1] |
Hi Jun Choe, Bataa Lkhagvasuren, Minsuk Yang. Wellposedness of the Keller-Segel Navier-Stokes equations in the critical Besov spaces. Communications on Pure and 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 and Continuous Dynamical Systems - B, 2016, 21 (4) : 1317-1327. doi: 10.3934/dcdsb.2016.21.1317 |
[3] |
Ling Liu, Jiashan Zheng, Gui Bao. Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3437-3460. doi: 10.3934/dcdsb.2020068 |
[4] |
Qi Wang. Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1231-1250. doi: 10.3934/dcdsb.2015.20.1231 |
[5] |
Guoqiang Ren, Bin Liu. Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3843-3883. doi: 10.3934/cpaa.2020170 |
[6] |
Tobias Black. Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 119-137. doi: 10.3934/dcdss.2020007 |
[7] |
Xinru Cao. Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1891-1904. doi: 10.3934/dcds.2015.35.1891 |
[8] |
Piotr Biler, Ignacio Guerra, Grzegorz Karch. Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2117-2126. doi: 10.3934/cpaa.2015.14.2117 |
[9] |
Hai-Yang Jin, Zhi-An Wang. Global stabilization of the full attraction-repulsion Keller-Segel system. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3509-3527. doi: 10.3934/dcds.2020027 |
[10] |
Hao Yu, Wei Wang, Sining Zheng. Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1635-1644. doi: 10.3934/dcdsb.2017078 |
[11] |
Luca Battaglia. A general existence result for stationary solutions to the Keller-Segel system. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 905-926. doi: 10.3934/dcds.2019038 |
[12] |
Grzegorz Karch, Maria E. Schonbek, Tomas P. Schonbek. Singularities of certain finite energy solutions to the Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 189-206. doi: 10.3934/dcds.2020008 |
[13] |
Peter Anthony, Sergey Zelik. Infinite-energy solutions for the Navier-Stokes equations in a strip revisited. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1361-1393. doi: 10.3934/cpaa.2014.13.1361 |
[14] |
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 and Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243 |
[15] |
Norikazu Saito. Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis. Communications on Pure and Applied Analysis, 2012, 11 (1) : 339-364. doi: 10.3934/cpaa.2012.11.339 |
[16] |
Zhichun Zhai. Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces. Communications on Pure and Applied Analysis, 2011, 10 (1) : 287-308. doi: 10.3934/cpaa.2011.10.287 |
[17] |
Minghua Yang, Zunwei Fu, Jinyi Sun. Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3427-3460. doi: 10.3934/dcdsb.2018284 |
[18] |
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 |
[19] |
Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085 |
[20] |
Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations and Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]