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February  2018, 11(1): 179-190. doi: 10.3934/krm.2018009

Global regularity for a model of Navier-Stokes equations with logarithmic sub-dissipation

Shuguang Shao 1,2,, , Shu Wang 1, and  Wen-Qing Xu 1,3,

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

* Corresponding author: Shuguang Shao

Received  October 2016 Revised  December 2016 Published  August 2017

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.

Keywords: Navier-Stokes equations, global regularity, sub-dissipation, energy estimates.
Mathematics Subject Classification: Primary: 35A01, 35B65; Secondary: 53C35.
Citation: Shuguang Shao, Shu Wang, Wen-Qing Xu. Global regularity for a model of Navier-Stokes equations with logarithmic sub-dissipation. Kinetic & Related Models, 2018, 11 (1) : 179-190. doi: 10.3934/krm.2018009
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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

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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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

T. Tao, Structure and Randomness: Pages from Year One of a Mathematical Blog American Mathematical Society, 2008. doi: 10.1090/mbk/059. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

T. Tao, Structure and Randomness: Pages from Year One of a Mathematical Blog American Mathematical Society, 2008. doi: 10.1090/mbk/059. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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