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Asymptotic behavior of the compressible non-isentropic Navier-Stokes-Maxwell system in $\mathbb{R}^3$
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Global regularity for a model of Navier-Stokes equations with logarithmic sub-dissipation
1. | College of Applied Sciences, Beijing University of Technology, Beijing 100124, China |
2. | School of Mathematics and Statistics, Nanyang Normal University, Nanyang 473061, China |
3. | Department of Mathematics and Statistics, California State University, Long Beach, Long Beach, CA 90840, USA |
In this paper, we study the global regularity to a three-dimensional logarithmic sub-dissipative Navier-Stokes model. This system takes the form of ${\partial _t}u +(\mathcal {D}^{-1/2}u)·\nabla u + \nabla p =-\mathcal {A}^2u$, where $\mathcal {D}$ and $\mathcal {A}$ are Fourier multipliers defined by $\mathcal {D}=|\nabla|$ and $\mathcal {A}= |\nabla|\ln^{-1/4}(e + λ \ln (e + |\nabla|)) $ with $λ≥q0$. The symbols of the $\mathcal {D}$ and $\mathcal {A}$ are $m(ξ) =\left| ξ \right|$ and $h(ξ) = \left| ξ \right| / g(ξ)$ respectively, where $g(ξ) = {\ln ^{{1 / 4}}}(e + λ \ln (e + |ξ|))$, $λ≥0$. It is clear that for the Navier-Stokes equations, global regularity is true under the assumption that $h(ξ) =|ξ|^α$ for $α≥q 5/4$. Here by changing the advection term we greatly weaken the dissipation to $ h(ξ)={{\left| ξ \right|} / g(ξ)}$. We prove the global well-posedness for any smooth initial data in $H^s(\mathbb{R}^3)$, $ s≥q3 $ by using the energy method.
References:
[1] |
H. Bahouri, J. Y. Chemin and R. Danchin,
Fourier Analysis and Nonlinear Partial Differential Equations Grundlehren Math. Wiss. (Fundamental Principles of Mathematical Sciences), vol. 343, Springer, Heidelberg, 2011, xvi+523pp.
doi: 10.1007/978-3-642-16830-7. |
[2] |
J.-Y. Chemin and I. Gallagher,
Wellposedness and stability results for the Navier-Stokes equations in $\mathbb{R}^3$, Ann. Inst. H. Poincar$\acute{e}$ Anal., Non Lin$\acute{e}$aire, 26 (2009), 599-624.
doi: 10.1016/j.anihpc.2007.05.008. |
[3] |
C. R. Doering and J. D. Gibbon,
Bounds on moments of the energy spectrum for weak solutions of the three-dimensional Navier-Stokes equations, Phys. D, 165 (2002), 163-175.
doi: 10.1016/S0167-2789(02)00427-X. |
[4] |
D. Fang and B. Han,
Global solution for the generalized anisotropic Navier-Stokes equations with large data, Mathematical Modeling and Analysis, 20 (2015), 205-231.
doi: 10.3846/13926292.2015.1020894. |
[5] |
C. L. Fefferman,
Existence and smoothness of the Navier-Stokes equation, in: J. Carlson, et al. (Eds.), The Millennium Prize Problems, Clay Math. Inst., (2006), 57-67.
|
[6] |
I. Gallagher and M. Paicu,
Remarks on the blow-up of solutions to a toy model for the Navier-Stokes equations, Proc. Amer. Math. Soc., 137 (2009), 2075-2083.
doi: 10.1090/S0002-9939-09-09765-2. |
[7] |
T. Y. Hou and Z. Lei,
On the stabilizing effect of convection in three-dimensional incompressible flows, Comm. Pure Appl. Math., 62 (2009), 501-564.
doi: 10.1002/cpa.20254. |
[8] |
T. Y. Hou, Z. Lei and C. M. Li,
Global regularity of the 3D axi-symmetric Navier-Stokes equations with anisotropic data, Comm. Partial Differential Equations, 33 (2008), 1622-1637.
doi: 10.1080/03605300802108057. |
[9] |
T. Y. Hou, Z. Lei, G. Luo, S. Wang and C. Zou,
On finite time singularity and global regularity of an axisymmetric model for the 3D Euler equations, Arch. Ration. Mech. Anal., 212 (2014), 683-706.
doi: 10.1007/s00205-013-0717-6. |
[10] |
T. Y. Hou and R. Li,
Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations, J. Nonlinear Sci., 16 (2006), 639-664.
doi: 10.1007/s00332-006-0800-3. |
[11] |
N. Katz and N. Pavlovi$\acute{c}$,
A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation, Geom. Funct. Anal., 12 (2002), 355-379.
doi: 10.1007/s00039-002-8250-z. |
[12] |
N. H. Katz and N. Pavlovic,
Finite time blow-up for a dyadic model of the Euler equations, Trans. Amer. Math. Soc., 357 (2005), 695-708.
doi: 10.1090/S0002-9947-04-03532-9. |
[13] |
H. Koch and D. Tataru,
Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[14] |
Z. Lei and F. H. Lin,
Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 1297-1304.
doi: 10.1002/cpa.20361. |
[15] |
Z. Lei, F. H. Lin and Y. Zhou,
Structure of helicity and global solutions of incompressible Navier-Stokes equation, Arch. Ration. Mech. Anal., 218 (2015), 1417-1430.
doi: 10.1007/s00205-015-0884-8. |
[16] |
Z. Lei, E. A. Navas and Q. S. Zhang,
A priori bound on the velocity in axially symmetric Navier-Stokes equations, Comm. Math. Phys., 341 (2016), 289-307.
doi: 10.1007/s00220-015-2496-4. |
[17] |
D. Li and Ya. Sinai,
Blow ups of complex solutions of the 3d-Navier-Stokes system and renormalization group method, J. Eur. Math. Soc.(JEMS), 10 (2008), 267-313.
doi: 10.4171/JEMS/111. |
[18] |
S. Montgomery-Smith,
Finite time blow up for a Navier-Stokes like equation, Proc. Amer. Math. Soc., 129 (2001), 3025-3029.
doi: 10.1090/S0002-9939-01-06062-2. |
[19] |
P. Plechac and V. Severak,
singular and regular solutions of a nonlinear parabolic system, Nonlinearity, 16 (2003), 2083-2097.
doi: 10.1088/0951-7715/16/6/313. |
[20] |
P. Plechac and V. Severak,
On self-similar singular solutions of the complex Ginzburg-Landau equation, Comm. Pure Appl. Math., 54 (2001), 1215-1242.
doi: 10.1002/cpa.3006. |
[21] |
T. Tao,
Localisation and compactness properties of the Navier-Stokes global regularity problem, Anal. PDE, 6 (2013), 25-107.
doi: 10.2140/apde.2013.6.25. |
[22] |
T. Tao,
Structure and Randomness: Pages from Year One of a Mathematical Blog American Mathematical Society, 2008.
doi: 10.1090/mbk/059. |
[23] |
T. Tao,
Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation, Anal. PDE, 2 (2009), 361-366.
doi: 10.2140/apde.2009.2.361. |
[24] |
T. Tao,
Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics 106 Conference Board of the Mathematical Sciences, Washington, DC, 2006.
doi: 10.1090/cbms/106. |
[25] |
T. Tao,
A quantitative formulation of the global regularity problem for the periodic Navier-Stokes equation, Dyn. Partial Differ Equ., 4 (2007), 293-302.
doi: 10.4310/DPDE.2007.v4.n4.a1. |
[26] |
K. Y. Wang,
Global regularity for a model of three-dimensional Navier-Stokes equation, J. Differential Equations, 258 (2015), 2969-2982.
doi: 10.1016/j.jde.2014.12.034. |
[27] |
Y. Zhou and Z. Lei,
Logarithmically improved criteria for Euler and Navier-Stokes equations, Commun. Pure Appl. Anal., 12 (2013), 2715-2719.
doi: 10.3934/cpaa.2013.12.2715. |
show all references
References:
[1] |
H. Bahouri, J. Y. Chemin and R. Danchin,
Fourier Analysis and Nonlinear Partial Differential Equations Grundlehren Math. Wiss. (Fundamental Principles of Mathematical Sciences), vol. 343, Springer, Heidelberg, 2011, xvi+523pp.
doi: 10.1007/978-3-642-16830-7. |
[2] |
J.-Y. Chemin and I. Gallagher,
Wellposedness and stability results for the Navier-Stokes equations in $\mathbb{R}^3$, Ann. Inst. H. Poincar$\acute{e}$ Anal., Non Lin$\acute{e}$aire, 26 (2009), 599-624.
doi: 10.1016/j.anihpc.2007.05.008. |
[3] |
C. R. Doering and J. D. Gibbon,
Bounds on moments of the energy spectrum for weak solutions of the three-dimensional Navier-Stokes equations, Phys. D, 165 (2002), 163-175.
doi: 10.1016/S0167-2789(02)00427-X. |
[4] |
D. Fang and B. Han,
Global solution for the generalized anisotropic Navier-Stokes equations with large data, Mathematical Modeling and Analysis, 20 (2015), 205-231.
doi: 10.3846/13926292.2015.1020894. |
[5] |
C. L. Fefferman,
Existence and smoothness of the Navier-Stokes equation, in: J. Carlson, et al. (Eds.), The Millennium Prize Problems, Clay Math. Inst., (2006), 57-67.
|
[6] |
I. Gallagher and M. Paicu,
Remarks on the blow-up of solutions to a toy model for the Navier-Stokes equations, Proc. Amer. Math. Soc., 137 (2009), 2075-2083.
doi: 10.1090/S0002-9939-09-09765-2. |
[7] |
T. Y. Hou and Z. Lei,
On the stabilizing effect of convection in three-dimensional incompressible flows, Comm. Pure Appl. Math., 62 (2009), 501-564.
doi: 10.1002/cpa.20254. |
[8] |
T. Y. Hou, Z. Lei and C. M. Li,
Global regularity of the 3D axi-symmetric Navier-Stokes equations with anisotropic data, Comm. Partial Differential Equations, 33 (2008), 1622-1637.
doi: 10.1080/03605300802108057. |
[9] |
T. Y. Hou, Z. Lei, G. Luo, S. Wang and C. Zou,
On finite time singularity and global regularity of an axisymmetric model for the 3D Euler equations, Arch. Ration. Mech. Anal., 212 (2014), 683-706.
doi: 10.1007/s00205-013-0717-6. |
[10] |
T. Y. Hou and R. Li,
Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations, J. Nonlinear Sci., 16 (2006), 639-664.
doi: 10.1007/s00332-006-0800-3. |
[11] |
N. Katz and N. Pavlovi$\acute{c}$,
A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation, Geom. Funct. Anal., 12 (2002), 355-379.
doi: 10.1007/s00039-002-8250-z. |
[12] |
N. H. Katz and N. Pavlovic,
Finite time blow-up for a dyadic model of the Euler equations, Trans. Amer. Math. Soc., 357 (2005), 695-708.
doi: 10.1090/S0002-9947-04-03532-9. |
[13] |
H. Koch and D. Tataru,
Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[14] |
Z. Lei and F. H. Lin,
Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 1297-1304.
doi: 10.1002/cpa.20361. |
[15] |
Z. Lei, F. H. Lin and Y. Zhou,
Structure of helicity and global solutions of incompressible Navier-Stokes equation, Arch. Ration. Mech. Anal., 218 (2015), 1417-1430.
doi: 10.1007/s00205-015-0884-8. |
[16] |
Z. Lei, E. A. Navas and Q. S. Zhang,
A priori bound on the velocity in axially symmetric Navier-Stokes equations, Comm. Math. Phys., 341 (2016), 289-307.
doi: 10.1007/s00220-015-2496-4. |
[17] |
D. Li and Ya. Sinai,
Blow ups of complex solutions of the 3d-Navier-Stokes system and renormalization group method, J. Eur. Math. Soc.(JEMS), 10 (2008), 267-313.
doi: 10.4171/JEMS/111. |
[18] |
S. Montgomery-Smith,
Finite time blow up for a Navier-Stokes like equation, Proc. Amer. Math. Soc., 129 (2001), 3025-3029.
doi: 10.1090/S0002-9939-01-06062-2. |
[19] |
P. Plechac and V. Severak,
singular and regular solutions of a nonlinear parabolic system, Nonlinearity, 16 (2003), 2083-2097.
doi: 10.1088/0951-7715/16/6/313. |
[20] |
P. Plechac and V. Severak,
On self-similar singular solutions of the complex Ginzburg-Landau equation, Comm. Pure Appl. Math., 54 (2001), 1215-1242.
doi: 10.1002/cpa.3006. |
[21] |
T. Tao,
Localisation and compactness properties of the Navier-Stokes global regularity problem, Anal. PDE, 6 (2013), 25-107.
doi: 10.2140/apde.2013.6.25. |
[22] |
T. Tao,
Structure and Randomness: Pages from Year One of a Mathematical Blog American Mathematical Society, 2008.
doi: 10.1090/mbk/059. |
[23] |
T. Tao,
Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation, Anal. PDE, 2 (2009), 361-366.
doi: 10.2140/apde.2009.2.361. |
[24] |
T. Tao,
Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics 106 Conference Board of the Mathematical Sciences, Washington, DC, 2006.
doi: 10.1090/cbms/106. |
[25] |
T. Tao,
A quantitative formulation of the global regularity problem for the periodic Navier-Stokes equation, Dyn. Partial Differ Equ., 4 (2007), 293-302.
doi: 10.4310/DPDE.2007.v4.n4.a1. |
[26] |
K. Y. Wang,
Global regularity for a model of three-dimensional Navier-Stokes equation, J. Differential Equations, 258 (2015), 2969-2982.
doi: 10.1016/j.jde.2014.12.034. |
[27] |
Y. Zhou and Z. Lei,
Logarithmically improved criteria for Euler and Navier-Stokes equations, Commun. Pure Appl. Anal., 12 (2013), 2715-2719.
doi: 10.3934/cpaa.2013.12.2715. |
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