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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

Jishan Fan 1, and  Kun Zhao 2,,

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  November 2018 Early access  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.

Keywords: Navier-Stokes-Keller-Segel system, initial-boundary value problem, classical solution, extensibility criteria, global well-posedness.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 35Q92.
Citation: Jishan Fan, Kun Zhao. Improved extensibility criteria and global well-posedness of a coupled chemotaxis-fluid model on bounded domains. Discrete and 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. 

[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.

[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. 

[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.

[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.

[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.

[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.

[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. 

[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. 

[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.

[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.

[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.

[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.

show all references

References:
[1]

H. Amann, Maximal regularity for nonautonomous evolution equations, Adv. Nonlinear Studies, 4 (2004), 417-430. 

[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.

[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. 

[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.

[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.

[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.

[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.

[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. 

[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. 

[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.

[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.

[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.

[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.

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