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Monotonicity and uniqueness of wave profiles for a three components lattice dynamical system
Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data
1. | Department of Mathematics, Sun Yat-sen University, Guangzhou, Guangdong 510275, China |
2. | Faculty of Information Technology, Macau University of Science and Technology, Macau |
In this paper, we mainly study the Cauchy problem of the Chemo-taxis-Navier-Stokes equations with initial data in critical Besov spaces. We first get the local wellposedness of the system in $\mathbb{R}^d \, (d≥2)$ by the Picard theorem, and then extend the local solutions to be global under the only smallness assumptions on $\|u_0^h\|_{\dot{B}_{p, 1}^{-1+\frac{d}{p}}}$, $\|n_0\|_{\dot{B}_{q, 1}^{-2+\frac{d}{q}}}$ and $\|c_0\|_{\dot{B}_{r, 1}^{\frac{d}{r}}}$. This obtained result implies the global wellposedness of the equations with large initial vertical velocity component. Moreover, by fully using the global wellposedness of the classical 2D Navier-Stokes equations and the weighted Chemin-Lerner space, we can also extend the obtained local solutions to be global in $\mathbb{R}^2$ provided the initial cell density $n_0$ and the initial chemical concentration $c_0$ are doubly exponential small compared with the initial velocity field $u_0$.
References:
[1] |
H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations Grundlehren Math. Wiss. , 343 Springer-Verlag, Berlin, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
J. M. Bony,
Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., 14 (1981), 209-246.
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[3] |
P. Biler and G. Karch,
Blow-up of solutions to generalized Keller-Segel model, J. Evol. Equ., 10 (2010), 247-262.
doi: 10.1007/s00028-009-0048-0. |
[4] |
A. Blanchet, J. Dolbeault and B. Perthame,
Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 44 (2006), 1-32.
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[5] |
V. Calvez and L. Corrias,
The parabolic-parabolic Keller-Segel model in $\mathbb{R}^2$, Commun. Math. Sci., 6 (2008), 417-447.
doi: 10.4310/CMS.2008.v6.n2.a8. |
[6] |
M. Chae, K. Kang and J. Lee,
Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297.
doi: 10.3934/dcds.2013.33.2271. |
[7] |
R. Danchin,
Local theory in critical spaces for compressible viscous and heat-conducting gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233.
|
[8] |
R. J. Duan, A. Lorz and P. A. Markowich,
Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673.
doi: 10.1080/03605302.2010.497199. |
[9] |
M. Di Francesco, A. Lorz and P. A. Markowich,
Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., 28 (2010), 1437-1453.
|
[10] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[11] |
J. Huang, M. Paicu and P. Zhang,
Global solutions to 2-D inhomogeneous Navier-Stokes system with general velocity, J. Math. Pures Appl., 100 (2013), 806-831.
doi: 10.1016/j.matpur.2013.03.003. |
[12] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[13] |
J. G. Liu and A. Lorz,
A coupled chemotaxis-fluid model: Global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652.
doi: 10.1016/j.anihpc.2011.04.005. |
[14] |
A. Lorz,
Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.
doi: 10.1142/S0218202510004507. |
[15] |
A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics 27, Cambridge University Press, Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511613203. |
[16] |
M. Paicu and P. Zhang,
Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces, Comm. Math. Phys., 307 (2011), 713-759.
doi: 10.1007/s00220-011-1350-6. |
[17] |
M. Paicu and P. Zhang,
Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262 (2012), 3556-3584.
doi: 10.1016/j.jfa.2012.01.022. |
[18] |
Y. Tao and M. Winkler,
Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 1901-1914.
|
[19] |
Y. Tao and M. Winkler,
Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.
doi: 10.1016/j.anihpc.2012.07.002. |
[20] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[21] |
Y. Tao and M. Winkler,
A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.
doi: 10.1137/100802943. |
[22] |
I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein,
Bacterial swimming and oxygen transport near constant lines, Proc. Natl. Acad. Sci., 102 (2005), 2277-2282.
|
[23] |
M. Winkler,
Global large-data solutions in a chemotaxis-Navier-Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.
doi: 10.1080/03605302.2011.591865. |
[24] |
M. Winkler,
Does a "volume-filling effect" always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.
doi: 10.1002/mma.1146. |
[25] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[26] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[27] |
C. Zhai and T. Zhang, Global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity, J. Math. Phys. , 56 (2015), 091512.
doi: 10.1063/1.4931467. |
show all references
References:
[1] |
H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations Grundlehren Math. Wiss. , 343 Springer-Verlag, Berlin, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
J. M. Bony,
Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., 14 (1981), 209-246.
|
[3] |
P. Biler and G. Karch,
Blow-up of solutions to generalized Keller-Segel model, J. Evol. Equ., 10 (2010), 247-262.
doi: 10.1007/s00028-009-0048-0. |
[4] |
A. Blanchet, J. Dolbeault and B. Perthame,
Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 44 (2006), 1-32.
|
[5] |
V. Calvez and L. Corrias,
The parabolic-parabolic Keller-Segel model in $\mathbb{R}^2$, Commun. Math. Sci., 6 (2008), 417-447.
doi: 10.4310/CMS.2008.v6.n2.a8. |
[6] |
M. Chae, K. Kang and J. Lee,
Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297.
doi: 10.3934/dcds.2013.33.2271. |
[7] |
R. Danchin,
Local theory in critical spaces for compressible viscous and heat-conducting gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233.
|
[8] |
R. J. Duan, A. Lorz and P. A. Markowich,
Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673.
doi: 10.1080/03605302.2010.497199. |
[9] |
M. Di Francesco, A. Lorz and P. A. Markowich,
Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., 28 (2010), 1437-1453.
|
[10] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[11] |
J. Huang, M. Paicu and P. Zhang,
Global solutions to 2-D inhomogeneous Navier-Stokes system with general velocity, J. Math. Pures Appl., 100 (2013), 806-831.
doi: 10.1016/j.matpur.2013.03.003. |
[12] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[13] |
J. G. Liu and A. Lorz,
A coupled chemotaxis-fluid model: Global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652.
doi: 10.1016/j.anihpc.2011.04.005. |
[14] |
A. Lorz,
Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.
doi: 10.1142/S0218202510004507. |
[15] |
A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics 27, Cambridge University Press, Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511613203. |
[16] |
M. Paicu and P. Zhang,
Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces, Comm. Math. Phys., 307 (2011), 713-759.
doi: 10.1007/s00220-011-1350-6. |
[17] |
M. Paicu and P. Zhang,
Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262 (2012), 3556-3584.
doi: 10.1016/j.jfa.2012.01.022. |
[18] |
Y. Tao and M. Winkler,
Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 1901-1914.
|
[19] |
Y. Tao and M. Winkler,
Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.
doi: 10.1016/j.anihpc.2012.07.002. |
[20] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[21] |
Y. Tao and M. Winkler,
A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.
doi: 10.1137/100802943. |
[22] |
I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein,
Bacterial swimming and oxygen transport near constant lines, Proc. Natl. Acad. Sci., 102 (2005), 2277-2282.
|
[23] |
M. Winkler,
Global large-data solutions in a chemotaxis-Navier-Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.
doi: 10.1080/03605302.2011.591865. |
[24] |
M. Winkler,
Does a "volume-filling effect" always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.
doi: 10.1002/mma.1146. |
[25] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[26] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[27] |
C. Zhai and T. Zhang, Global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity, J. Math. Phys. , 56 (2015), 091512.
doi: 10.1063/1.4931467. |
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