# American Institute of Mathematical Sciences

November  2018, 23(9): 3949-3967. doi: 10.3934/dcdsb.2018119

## Improved extensibility criteria and global well-posedness of a coupled chemotaxis-fluid model on bounded domains

 1 Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China 2 Department of Mathematics, Tulane University, New Orleans, LA 70118, USA

* Corresponding author: Kun Zhao

Received  May 2017 Published  April 2018

This paper is contributed to the qualitative analysis of a coupled chemotaxis-fluid model on bounded domains in multiple spatial dimensions. Based on scaling-invariant argument and energy method, several optimal extensibility criteria for local classical solutions are established. As a by-product, a global well-posedness result is obtained in the two-dimensional case for general initial data.

Citation: Jishan Fan, Kun Zhao. Improved extensibility criteria and global well-posedness of a coupled chemotaxis-fluid model on bounded domains. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3949-3967. doi: 10.3934/dcdsb.2018119
##### References:
 [1] H. Amann, Maximal regularity for nonautonomous evolution equations, Adv. Nonlinear Studies, 4 (2004), 417-430. Google Scholar [2] J. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., 94 (1984), 61-66. doi: 10.1007/BF01212349. Google Scholar [3] M. Chae, K. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Disc. Cont. Dyn. Syst. A, 33 (2013), 2271-2297. Google Scholar [4] M. Chae, K. Kang and J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differential Equations, 39 (2014), 1205-1235. doi: 10.1080/03605302.2013.852224. Google Scholar [5] A. Chertock, K. Fellner, A. Kurganov, A. Lorz and P. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach, J. Fluid Mechanics, 694 (2012), 155-190. doi: 10.1017/jfm.2011.534. Google Scholar [6] A. Ferrari, On the blow-up of solutions of 3-D Euler equations in a bounded domain, Comm. Math. Phys., 155 (1993), 277-294. doi: 10.1007/BF02097394. Google Scholar [7] T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. Google Scholar [8] A. Hillesdon, T. Pedley and O. Kessler, The development of concentration gradients in a suspension of chemotactic bacteria, Bull. Math. Biol., 57 (1995), 299-344. Google Scholar [9] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences Ⅰ, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. Google Scholar [10] H. Kozono, T. Ogawa and Y. Taniuchi, Navier-Stokes equations in the Besov space near $L^∞$ and BMO, Kyushu J. Math., 57 (2003), 303-324. doi: 10.2206/kyushujm.57.303. Google Scholar [11] H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semilinear evolution equations, Math. Z., 242 (2002), 251-278. doi: 10.1007/s002090100332. Google Scholar [12] H. Kozono and Y. Shimada, Bilinear estimates in homogeneous Triebel-Lizorkin spaces and the Navier-Stokes equations, Math. Nachr., 276 (2004), 63-74. doi: 10.1002/mana.200310213. Google Scholar [13] I. Tuval, L. Cisneros, C. Dombrowski, C. Wolgemuth, J. Kessler and R. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. National Academy Sciences -U.S.A., 102 (2005), 2277-2282. doi: 10.1073/pnas.0406724102. Google Scholar

show all references

##### References:
 [1] H. Amann, Maximal regularity for nonautonomous evolution equations, Adv. Nonlinear Studies, 4 (2004), 417-430. Google Scholar [2] J. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., 94 (1984), 61-66. doi: 10.1007/BF01212349. Google Scholar [3] M. Chae, K. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Disc. Cont. Dyn. Syst. A, 33 (2013), 2271-2297. Google Scholar [4] M. Chae, K. Kang and J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differential Equations, 39 (2014), 1205-1235. doi: 10.1080/03605302.2013.852224. Google Scholar [5] A. Chertock, K. Fellner, A. Kurganov, A. Lorz and P. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach, J. Fluid Mechanics, 694 (2012), 155-190. doi: 10.1017/jfm.2011.534. Google Scholar [6] A. Ferrari, On the blow-up of solutions of 3-D Euler equations in a bounded domain, Comm. Math. Phys., 155 (1993), 277-294. doi: 10.1007/BF02097394. Google Scholar [7] T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. Google Scholar [8] A. Hillesdon, T. Pedley and O. Kessler, The development of concentration gradients in a suspension of chemotactic bacteria, Bull. Math. Biol., 57 (1995), 299-344. Google Scholar [9] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences Ⅰ, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. Google Scholar [10] H. Kozono, T. Ogawa and Y. Taniuchi, Navier-Stokes equations in the Besov space near $L^∞$ and BMO, Kyushu J. Math., 57 (2003), 303-324. doi: 10.2206/kyushujm.57.303. Google Scholar [11] H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semilinear evolution equations, Math. Z., 242 (2002), 251-278. doi: 10.1007/s002090100332. Google Scholar [12] H. Kozono and Y. Shimada, Bilinear estimates in homogeneous Triebel-Lizorkin spaces and the Navier-Stokes equations, Math. Nachr., 276 (2004), 63-74. doi: 10.1002/mana.200310213. Google Scholar [13] I. Tuval, L. Cisneros, C. Dombrowski, C. Wolgemuth, J. Kessler and R. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. National Academy Sciences -U.S.A., 102 (2005), 2277-2282. doi: 10.1073/pnas.0406724102. Google Scholar
 [1] Tatsien Li, Libin Wang. Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 59-78. doi: 10.3934/dcds.2005.12.59 [2] Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135 [3] Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61 [4] Xiaoyun Cai, Liangwen Liao, Yongzhong Sun. Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 917-923. doi: 10.3934/dcdss.2014.7.917 [5] Peng Jiang. Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3015-3037. doi: 10.3934/dcds.2015.35.3015 [6] Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D Navier-Stokes-Maxwell system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5817-5835. doi: 10.3934/dcds.2016056 [7] 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 [8] Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319 [9] Elena Rossi. Well-posedness of general 1D initial boundary value problems for scalar balance laws. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3577-3608. doi: 10.3934/dcds.2019147 [10] Gilles Carbou, Bernard Hanouzet. Relaxation approximation of the Kerr model for the impedance initial-boundary value problem. Conference Publications, 2007, 2007 (Special) : 212-220. doi: 10.3934/proc.2007.2007.212 [11] Xianpeng Hu, Dehua Wang. The initial-boundary value problem for the compressible viscoelastic flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 917-934. doi: 10.3934/dcds.2015.35.917 [12] Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101 [13] Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35 [14] Weimin Peng, Yi Zhou. Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3845-3856. doi: 10.3934/dcds.2016.36.3845 [15] Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148 [16] Jishan Fan, Yueling Jia. Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum. Kinetic & Related Models, 2018, 11 (1) : 97-106. doi: 10.3934/krm.2018005 [17] Yuning Liu, Wei Wang. On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3879-3899. doi: 10.3934/dcdsb.2018115 [18] Hui Huang, Jian-Guo Liu. Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos. Kinetic & Related Models, 2016, 9 (4) : 715-748. doi: 10.3934/krm.2016013 [19] Daoyuan Fang, Ruizhao Zi. On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3517-3541. doi: 10.3934/dcds.2013.33.3517 [20] Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations & Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195

2018 Impact Factor: 1.008