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Asymptotic behavior of the compressible non-isentropic Navier-Stokes-Maxwell system in $\mathbb{R}^3$
February 2018, 11(1): 179-190. doi: 10.3934/krm.2018009

## 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

* 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.

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. [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. 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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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##### 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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