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
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H. KozonoT. 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

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H. KozonoT. 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

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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

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show all references

References:
[1]

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

[2]

J. BealeT. 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. ChaeK. 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. ChaeK. 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. ChertockK. FellnerA. KurganovA. 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. HillesdonT. 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. KozonoT. 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. KozonoT. 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. TuvalL. CisnerosC. DombrowskiC. WolgemuthJ. 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

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