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Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain
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 |
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.
References:
[1] |
H. Amann,
Maximal regularity for nonautonomous evolution equations, Adv. Nonlinear Studies, 4 (2004), 417-430.
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[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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