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

Jishan Fan; Kun Zhao

doi:10.3934/dcdsb.2018119

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2018, Volume 23, Issue 9: 3949-3967. Doi: 10.3934/dcdsb.2018119

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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
* Corresponding author: Kun Zhao

Received: April 30, 2017

Early access: April 2018

Published: September 2018

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q35; Secondary: 35Q92.

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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.