# American Institute of Mathematical Sciences

• Previous Article
Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source
• CPAA Home
• This Issue
• Next Article
Regularity and nonexistence of solutions for a system involving the fractional Laplacian
November  2015, 14(6): 2453-2464. doi: 10.3934/cpaa.2015.14.2453

## Wellposedness of the Keller-Segel Navier-Stokes equations in the critical Besov spaces

 1 Department of Mathematics, Yonsei University, 120-749 SeoDaeMun-gu, Seoul, South Korea 2 Mathematics Department,Yonsei University, Seoul 120-749, South Korea 3 School of Mathematics, Korea Institute for Advanced Study, Seoul 130-722, South Korea

Received  April 2015 Revised  July 2015 Published  September 2015

We consider the Keller-Segel model coupled with the incompressible Navier-Stokes equations in dimension three. We prove the local in time existence of the solution for large initial data and the global in time existence of the solution for small initial data plus some smallness condition on the gravitational potential in the critical Besov spaces, which are new results for the model.
Citation: 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
##### References:
 [1] H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],, Springer, (2011). doi: 10.1007/978-3-642-16830-7. Google Scholar [2] M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations,, in \emph{Handbook of Mathematical Fluid Dynamics. Vol. III}, (2004), 161. Google Scholar [3] M. Chae, K. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 2271. doi: 10.3934/dcds.2013.33.2271. Google Scholar [4] R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations,, \emph{Comm. Partial Differential Equations}, 35 (2010), 1635. doi: 10.1080/03605302.2010.497199. Google Scholar [5] H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I,, \emph{Arch. Rational Mech. Anal.}, 16 (1964), 269. Google Scholar [6] P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, vol. 431 of Chapman & Hall/CRC Research Notes in Mathematics,, Chapman & Hall/CRC, (2002). doi: 10.1201/9781420035674. Google Scholar [7] A. Lorz, Coupled chemotaxis fluid model,, \emph{Math. Models Methods Appl. Sci.}, 20 (2010), 987. doi: 10.1142/S0218202510004507. Google Scholar [8] I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines,, \emph{Proceedings of the National Academy of Sciences of the United States of America}, 102 (2005), 2277. Google Scholar [9] Q. Zhang, Local well-posedness for the chemotaxis-Navier-Stokes equations in Besov spaces,, \emph{Nonlinear Anal. Real World Appl.}, 17 (2014), 89. doi: 10.1016/j.nonrwa.2013.10.008. Google Scholar

show all references

##### References:
 [1] H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],, Springer, (2011). doi: 10.1007/978-3-642-16830-7. Google Scholar [2] M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations,, in \emph{Handbook of Mathematical Fluid Dynamics. Vol. III}, (2004), 161. Google Scholar [3] M. Chae, K. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 2271. doi: 10.3934/dcds.2013.33.2271. Google Scholar [4] R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations,, \emph{Comm. Partial Differential Equations}, 35 (2010), 1635. doi: 10.1080/03605302.2010.497199. Google Scholar [5] H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I,, \emph{Arch. Rational Mech. Anal.}, 16 (1964), 269. Google Scholar [6] P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, vol. 431 of Chapman & Hall/CRC Research Notes in Mathematics,, Chapman & Hall/CRC, (2002). doi: 10.1201/9781420035674. Google Scholar [7] A. Lorz, Coupled chemotaxis fluid model,, \emph{Math. Models Methods Appl. Sci.}, 20 (2010), 987. doi: 10.1142/S0218202510004507. Google Scholar [8] I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines,, \emph{Proceedings of the National Academy of Sciences of the United States of America}, 102 (2005), 2277. Google Scholar [9] Q. Zhang, Local well-posedness for the chemotaxis-Navier-Stokes equations in Besov spaces,, \emph{Nonlinear Anal. Real World Appl.}, 17 (2014), 89. doi: 10.1016/j.nonrwa.2013.10.008. Google Scholar
 [1] 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 [2] 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 [3] Daoyuan Fang, Chenyin Qian. Regularity criterion for 3D Navier-Stokes equations in Besov spaces. Communications on Pure & Applied Analysis, 2014, 13 (2) : 585-603. doi: 10.3934/cpaa.2014.13.585 [4] Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319 [5] Wenting Cong, Jian-Guo Liu. A degenerate $p$-Laplacian Keller-Segel model. Kinetic & Related Models, 2016, 9 (4) : 687-714. doi: 10.3934/krm.2016012 [6] 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 [7] Minghua Yang, Jinyi Sun. Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1617-1639. doi: 10.3934/cpaa.2017078 [8] Xue-Li Song, Yan-Ren Hou. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 239-252. doi: 10.3934/dcds.2011.31.239 [9] Hi Jun Choe, Hyea Hyun Kim, Do Wan Kim, Yongsik Kim. Meshless method for the stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 495-526. doi: 10.3934/dcdsb.2001.1.495 [10] Keyan Wang. On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1067-1072. doi: 10.3934/cpaa.2009.8.1067 [11] Hi Jun Choe, Do Wan Kim, Yongsik Kim. Meshfree method for the non-stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 17-39. doi: 10.3934/dcdsb.2006.6.17 [12] Roberta Bianchini, Roberto Natalini. Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations. Kinetic & Related Models, 2019, 12 (1) : 133-158. doi: 10.3934/krm.2019006 [13] Xiaoping Zhai, Yongsheng Li, Wei Yan. Global well-posedness for the 3-D incompressible MHD equations in the critical Besov spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1865-1884. doi: 10.3934/cpaa.2015.14.1865 [14] Shangbing Ai, Zhian Wang. Traveling bands for the Keller-Segel model with population growth. Mathematical Biosciences & Engineering, 2015, 12 (4) : 717-737. doi: 10.3934/mbe.2015.12.717 [15] Vincent Calvez, Benoȋt Perthame, Shugo Yasuda. Traveling wave and aggregation in a flux-limited Keller-Segel model. Kinetic & Related Models, 2018, 11 (4) : 891-909. doi: 10.3934/krm.2018035 [16] Yajing Zhang, Xinfu Chen, Jianghao Hao, Xin Lai, Cong Qin. Dynamics of spike in a Keller-Segel's minimal chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1109-1127. doi: 10.3934/dcds.2017046 [17] Xie Li, Zhaoyin Xiang. Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3503-3531. doi: 10.3934/dcds.2015.35.3503 [18] Jan Burczak, Rafael Granero-Belinchón. Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 139-164. doi: 10.3934/dcdss.2020008 [19] Jie Jiang. Global stability of Keller–Segel systems in critical Lebesgue spaces. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 609-634. doi: 10.3934/dcds.2020025 [20] Linjie Xiong. Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary. Kinetic & Related Models, 2018, 11 (3) : 469-490. doi: 10.3934/krm.2018021

2018 Impact Factor: 0.925