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Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces
1. | Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang, 330032, China |
2. | Department of Mathematics, Linyi University, Linyi, 276000, China |
3. | School of Mathematical Sciences, Qufu Normal University, Qufu, 273100, China |
4. | Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China |
$(u_{0}, n_{0}, c_{0})$ |
$C_{0}$ |
$\phi \in \dot B_{p,1}^{3/p}({\mathbb{R}^3})$ |
$(u_{0}, n_{0}, c_{0}): = (u_{0}^{1}, u_{0}^{2}, u_{0}^{3}, n_{0}, c_{0}): = (u_{0}^{h}, u_{0}^{3}, n_{0}, c_{0})$ |
$\begin{equation*}\begin{aligned} &\left(\left\|u_{0}^{h}\right\|_{\dot{B}^{-1+3/p}_{p, 1}(\mathbb{R}^3)}+\left\|\left(n_{0}, c_{0}\right)\right\|_{\dot{B}^{-2+3/q}_{q, 1}(\mathbb{R}^3) \times \dot{B}^{3/q}_{q, 1}(\mathbb{R}^3)}\right)\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times\exp\left\{C_{0}\left(\left\|u_{0}^{3}\right\|_{\dot{B}^{-1+3/p}_{p, 1}(\mathbb{R}^3)}+1\right)^{2}\right\} \leq \sigma_{0}\end{aligned}\end{equation*}$ |
$p, q$ |
$1<p, q<6,\frac{1}{p}+\frac{1}{q}>\frac{2}{3}$ |
$\frac{1}{\min\{p, q\}}-\frac{1}{\max\{p, q\}} \le \frac{1}{3}$ |
$u_{0}^{3}$ |
References:
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M. Chae, K. Kang and J. Lee,
Existence of smooth solutions to coupled chemotaxis-fuid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297.
doi: 10.3934/dcds.2013.33.2271. |
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M. Chae, K. Kang and J. Lee,
Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differ. Equ., 39 (2014), 1205-1235.
doi: 10.1080/03605302.2013.852224. |
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M. Chae, K. Kang and J. Lee, Asymptotic behaviors of solutions for an aerobatic model coupled to fluid equations, J. Korean Math. Soc., 53 (2016), 127-146, arxiv.org/abs/1403.3713
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Théorémes dunicité pour le systéme de Navier-Stokes tridimensionnal, J. Anal. Math., 77 (1999), 27-50.
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Y. Chemin, M. Paicu and P. Zhang,
Global large solutions to 3-D inhomogeneous Navier-Stokes system with one slow variable diffusion, J. Differential Equations, 256 (2014), 223-252.
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Y. Chemin and P. Zhang,
On the global well-posedness to the 3D incompressible anisotropic Navier-Stokes equations, Comm. Math. Phys., 272 (2007), 529-566.
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H. Choe and B. Lkhagvasuren,
Global existence result for Chemotaxis Navier-Stokes equations in the critical Besov spaces, J. Math. Anal. Appl., 446 (2017), 1415-1426.
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Local theory in critical spaces for compressible viscous and heat-conducting gases, Comm. Partial Differ. Equ., 26 (2001), 1183-1233.
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Global solutions to the coupled chemotaxis-fuid equations, Comm. Partial Differ. Equ., 35 (2010), 1635-1673.
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Global solutions to the coupled chemotaxis-fuid equations, Int. Math. Res. Not. IMRN, (2014), 1833-1852.
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Reaction terms avoiding aggregation in slow fluids, Nonlinear Anal. Real World Appl., 21 (2015), 110-126.
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Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., 28 (2010), 1437-1453.
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A blow-up mechanism for chemotaxis model, Ann. Sc. Norm. Super. Pisa., 24 (1997), 633-683.
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Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.
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Global well-posedness of incompressible inhomogeneous fluid systems with bounded density or non-Lipschitz velocity, Arch. Ration. Mech. Anal., 209 (2013), 631-382.
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Strong Lp-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.
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E. Keller and L. Segel,
Initiation of slide mold aggregation viewd as an instability, J. Theor. Biol., 6 (1970), 399-415.
|
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E. Keller and L. Segel,
Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.
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Enhancement of the traveling front speeds in reaction-diffusion equations with advection, Annales de l'Institut Henri Poincaré(C) Non Linear Analysis, 18 (2001), 309-358.
doi: 10.1016/S0294-1449(01)00068-3. |
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Suppression of chemotactic explosion by mixing, Arch. Ration. Mech. Anal., 222 (2016), 1077-1112.
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J. Leray,
Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.
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H. Koch and D. Tataru,
Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.
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H. Kozono and M. Nakao,
Periodic solutions of the Navier-Stokes equations in unbounded domains, Tohoku Math. J., 48 (1996), 33-50.
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Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differ. Equ., 19 (1994), 959-1014.
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Global well-posedness and long time decay of the 3D Boussinesq equations, J. Differential Equations, 263 (2017), 8649-8665.
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J. Liu and A. Lorz,
A coupled chemotaxis-fluid model: Global existence, Ann. Inst. H.Poincaré Anal. Non Linéaire, 28 (2011), 643-652.
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Q. Liu, T. Zhang and J. Zhao,
Global solutions to the 3D incompressible nematic liquid crystal system, J. Differential Equations, 258 (2015), 1519-1547.
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A. Lorz,
Coupled chemotaxis fuid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.
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A coupled Keller-Segel-Stokes model: Global existence for small initial data and blow-up delay, Commun. Math. Sci., 10 (2012), 555-574.
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équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoam., 21 (2005), 179-235.
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Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.
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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.
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show all references
References:
[1] |
M. Chae, K. Kang and J. Lee,
Existence of smooth solutions to coupled chemotaxis-fuid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297.
doi: 10.3934/dcds.2013.33.2271. |
[2] |
M. Chae, K. Kang and J. Lee,
Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differ. Equ., 39 (2014), 1205-1235.
doi: 10.1080/03605302.2013.852224. |
[3] |
M. Chae, K. Kang and J. Lee, Asymptotic behaviors of solutions for an aerobatic model coupled to fluid equations, J. Korean Math. Soc., 53 (2016), 127-146, arxiv.org/abs/1403.3713
doi: 10.4134/JKMS.2016.53.1.127. |
[4] |
Y. Chemin,
Théorémes dunicité pour le systéme de Navier-Stokes tridimensionnal, J. Anal. Math., 77 (1999), 27-50.
doi: 10.1007/BF02791256. |
[5] |
Y. Chemin, M. Paicu and P. Zhang,
Global large solutions to 3-D inhomogeneous Navier-Stokes system with one slow variable diffusion, J. Differential Equations, 256 (2014), 223-252.
doi: 10.1016/j.jde.2013.09.004. |
[6] |
Y. Chemin and P. Zhang,
On the global well-posedness to the 3D incompressible anisotropic Navier-Stokes equations, Comm. Math. Phys., 272 (2007), 529-566.
doi: 10.1007/s00220-007-0236-0. |
[7] |
H. Choe and B. Lkhagvasuren,
Global existence result for Chemotaxis Navier-Stokes equations in the critical Besov spaces, J. Math. Anal. Appl., 446 (2017), 1415-1426.
doi: 10.1016/j.jmaa.2016.09.050. |
[8] |
Y. Chung and K. Kang, Existence of global solutions for a Keller-Segel-fluid equations with nonlinear diffusion, arXiv: 1504.02274. |
[9] |
P. Constantin, A. Kiselev, L. Ryzhik and A. Zlato |
[10] |
R. Danchin, Fourier Analysis Methods for PDEs, Lecture Notes, 14, November 2005. |
[11] |
R. Danchin,
Local theory in critical spaces for compressible viscous and heat-conducting gases, Comm. Partial Differ. Equ., 26 (2001), 1183-1233.
doi: 10.1081/PDE-100106132. |
[12] |
R. Duan, A. Lorz and P. Markowich,
Global solutions to the coupled chemotaxis-fuid equations, Comm. Partial Differ. Equ., 35 (2010), 1635-1673.
doi: 10.1080/03605302.2010.497199. |
[13] |
R. Duan and Z. Xiang,
Global solutions to the coupled chemotaxis-fuid equations, Int. Math. Res. Not. IMRN, (2014), 1833-1852.
doi: 10.1093/imrn/rns270. |
[14] |
E. Espejo and T. Suzuki,
Reaction terms avoiding aggregation in slow fluids, Nonlinear Anal. Real World Appl., 21 (2015), 110-126.
doi: 10.1016/j.nonrwa.2014.07.001. |
[15] |
M. Francesco, A. Lorz and P. Markowich,
Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., 28 (2010), 1437-1453.
doi: 10.3934/dcds.2010.28.1437. |
[16] |
H. Fujita and T. Kato,
On the Navier-Stokes initial value problem I, Arch. Ration. Mech. Anal., 16 (1964), 269-315.
doi: 10.1007/BF00276188. |
[17] |
B. Hajer, Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Berlin, 2011.
doi: 10.1007/978-3-642-16830-7. |
[18] |
M. Herrero and L. Velazquez,
A blow-up mechanism for chemotaxis model, Ann. Sc. Norm. Super. Pisa., 24 (1997), 633-683.
|
[19] |
D. Horstman and G. Wang,
Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[20] |
J. Huang, M. Paicu and P. Zhang,
Global well-posedness of incompressible inhomogeneous fluid systems with bounded density or non-Lipschitz velocity, Arch. Ration. Mech. Anal., 209 (2013), 631-382.
doi: 10.1007/s00205-013-0624-x. |
[21] |
T. Kato,
Strong Lp-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[22] |
E. Keller and L. Segel,
Initiation of slide mold aggregation viewd as an instability, J. Theor. Biol., 6 (1970), 399-415.
|
[23] |
E. Keller and L. Segel,
Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[24] |
A. Kiselev and L. Ryzhik,
Enhancement of the traveling front speeds in reaction-diffusion equations with advection, Annales de l'Institut Henri Poincaré(C) Non Linear Analysis, 18 (2001), 309-358.
doi: 10.1016/S0294-1449(01)00068-3. |
[25] |
A. Kiselev and X. Xu,
Suppression of chemotactic explosion by mixing, Arch. Ration. Mech. Anal., 222 (2016), 1077-1112.
doi: 10.1007/s00205-016-1017-8. |
[26] |
J. Leray,
Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
[27] |
H. Koch and D. Tataru,
Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[28] |
H. Kozono and M. Nakao,
Periodic solutions of the Navier-Stokes equations in unbounded domains, Tohoku Math. J., 48 (1996), 33-50.
doi: 10.2748/tmj/1178225411. |
[29] |
H. Kozono and M. Yamazaki,
Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differ. Equ., 19 (1994), 959-1014.
doi: 10.1080/03605309408821042. |
[30] |
H. Liu and H. Gao,
Global well-posedness and long time decay of the 3D Boussinesq equations, J. Differential Equations, 263 (2017), 8649-8665.
doi: 10.1016/j.jde.2017.08.049. |
[31] |
J. 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. |
[32] |
Q. Liu, T. Zhang and J. Zhao,
Global solutions to the 3D incompressible nematic liquid crystal system, J. Differential Equations, 258 (2015), 1519-1547.
doi: 10.1016/j.jde.2014.11.002. |
[33] |
A. Lorz,
Coupled chemotaxis fuid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.
doi: 10.1142/S0218202510004507. |
[34] |
A. Lorz,
A coupled Keller-Segel-Stokes model: Global existence for small initial data and blow-up delay, Commun. Math. Sci., 10 (2012), 555-574.
doi: 10.4310/CMS.2012.v10.n2.a7. |
[35] |
Y. Minsuk, L. Bataa and C. Hi,
Well posedness of the Keller-Segel Navier-Stokes equations in the critical Besov spaces, Commun. Pure Appl. Anal., 14 (2015), 2453-2464.
doi: 10.3934/cpaa.2015.14.2453. |
[36] |
T. Nagai, T. Senba and K. Yoshida,
Applications of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial Ekvac., 40 (1997), 411-433.
|
[37] |
K. Osaki and A. Yagi,
Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial Ekvac., 44 (2001), 441-469.
|
[38] |
M. Paicu,
équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoam., 21 (2005), 179-235.
doi: 10.4171/RMI/420. |
[39] |
M. Paicu and P. Zhang,
Global solutions to the 3D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262 (2012), 3556-3584.
doi: 10.1016/j.jfa.2012.01.022. |
[40] |
C. Patlak,
Random walk with persistence and external bias, Bull. Math. Biol. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[41] |
F. Planchon,
Sur un inégalité de type Poincaré, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 21-23.
doi: 10.1016/S0764-4442(00)88138-0. |
[42] |
Y. Tao,
Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.
doi: 10.1016/j.jmaa.2011.02.041. |
[43] |
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.
doi: 10.3934/dcds.2012.32.1901. |
[44] |
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.
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