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May  2013, 12(3): 1299-1306. doi: 10.3934/cpaa.2013.12.1299

An anisotropic regularity criterion for the 3D Navier-Stokes equations

Xuanji Jia 1, and  Zaihong Jiang 2,

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, P. R.
2. 

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China

Received  January 2012 Revised  June 2012 Published  September 2012

In this paper, we establish an anisotropic regularity criterion for the 3D incompressible Navier-Stokes equations. It is proved that a weak solution $u$ is regular on $[0,T]$, provided $\frac{\partial u_3}{\partial x_3} \in L^{t_1}(0,T;L^{s_1}(R^3))$, with $\frac{2}{t_1}+\frac{3}{s_1}\leq 2$, $s_1\in(\frac{3}{2},+\infty]$ and $\nabla_h u_3 \in L^{t_2}(0, T; L^{s_2}(R^3))$, with either $\frac{2}{t_2}+\frac{3}{s_2}\leq \frac{19}{12}+\frac{1}{2s_2}$, $s_2\in(\frac{30}{19},3]$ or $ \frac{2}{t_2}+\frac{3}{s_2}\leq \frac{3}{2}+\frac{3}{4s_2}$, $s_2\in(3,+\infty]$. Our result in fact improves a regularity criterion of Zhou and Pokorný [Nonlinearity 23 (2010), 1097--1107].
Keywords: anisotropic regularity criterion., Navier-Stokes equations.
Mathematics Subject Classification: Primary: 35Q35, 35B65; Secondary: 76D0.
Citation: Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299
References:
[1]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.  Google Scholar

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E. Hopf, Über die Anfangwertaufgaben für die hydromischen Grundgleichungen, Math. Nachr., 4 (1951), 213-321. doi: 10.1002/mana.3210040121.  Google Scholar

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G. Prodi, Un teorema di unicità per el equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182. doi: 10.1007/BF02410664.  Google Scholar

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J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rat. Mech. Anal., 9 (1962), 187-195. doi: 10.1007/BF00253344.  Google Scholar

[5]

L. Escauriaza, G. Seregin and V. Šverák, Backward uniqueness for parabolic equations, Arch. Rat. Mech. Anal., 169 (2003), 147-157. doi: 10.1007/s00205-003-0263-8.  Google Scholar

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H. Beirão da Veiga, A new regularity class for the Navier-stokes equations in $\mathbfR^n$, Chin. Ann. Math., 16 (1995), 407-412.  Google Scholar

[7]

J. Neustupa, A. Novotný and P. Penel, An interior regularity of a weak solution to the Navier-Stokes equations in dependence on one component of velocity, in "Topics in mathematical fluid mechanics," (2002), 163-183.  Google Scholar

[8]

Y. Zhou, A new regularity criterion for weak solutions to the Navier-Stokes equations, J. Math. Pures Appl., 84 (2005), 1496-1514. doi: 10.1016/j.matpur.2005.07.003.  Google Scholar

[9]

Y. Zhou, A new regularity result for the Navier-Stokes equations in terms of the gradient of one velocity component, Methods Appl. Anal., 9 (2002), 563-578.  Google Scholar

[10]

M. Pokorný, On the result of He concerning the smoothness of solutions to the Navier-Stokes equations, Electron. J. Diff. Eqns., 11 (2003), 1-8.  Google Scholar

[11]

C. Cao and E. S. Titi, Regularity criteria for the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57 (2008), 2643-2661. doi: 10.1512/iumj.2008.57.3719.  Google Scholar

[12]

I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 453-469. doi: 10.1088/0951-7715/19/2/012.  Google Scholar

[13]

I. Kukavica and M. Ziane, Navier-Stokes equations with regularity in one direction, J. Math. Phys., 48 (2007), 065203. doi: 10.1063/1.2395919.  Google Scholar

[14]

Y. Zhou and M. Pokorný, On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component, J. Math. Phys., 50 (2009), 123514. doi: 10.1063/1.3268589.  Google Scholar

[15]

Y. Zhou and M. Pokorný, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107. doi: 10.1088/0951-7715/23/5/004.  Google Scholar

show all references

References:
[1]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.  Google Scholar

[2]

E. Hopf, Über die Anfangwertaufgaben für die hydromischen Grundgleichungen, Math. Nachr., 4 (1951), 213-321. doi: 10.1002/mana.3210040121.  Google Scholar

[3]

G. Prodi, Un teorema di unicità per el equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182. doi: 10.1007/BF02410664.  Google Scholar

[4]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rat. Mech. Anal., 9 (1962), 187-195. doi: 10.1007/BF00253344.  Google Scholar

[5]

L. Escauriaza, G. Seregin and V. Šverák, Backward uniqueness for parabolic equations, Arch. Rat. Mech. Anal., 169 (2003), 147-157. doi: 10.1007/s00205-003-0263-8.  Google Scholar

[6]

H. Beirão da Veiga, A new regularity class for the Navier-stokes equations in $\mathbfR^n$, Chin. Ann. Math., 16 (1995), 407-412.  Google Scholar

[7]

J. Neustupa, A. Novotný and P. Penel, An interior regularity of a weak solution to the Navier-Stokes equations in dependence on one component of velocity, in "Topics in mathematical fluid mechanics," (2002), 163-183.  Google Scholar

[8]

Y. Zhou, A new regularity criterion for weak solutions to the Navier-Stokes equations, J. Math. Pures Appl., 84 (2005), 1496-1514. doi: 10.1016/j.matpur.2005.07.003.  Google Scholar

[9]

Y. Zhou, A new regularity result for the Navier-Stokes equations in terms of the gradient of one velocity component, Methods Appl. Anal., 9 (2002), 563-578.  Google Scholar

[10]

M. Pokorný, On the result of He concerning the smoothness of solutions to the Navier-Stokes equations, Electron. J. Diff. Eqns., 11 (2003), 1-8.  Google Scholar

[11]

C. Cao and E. S. Titi, Regularity criteria for the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57 (2008), 2643-2661. doi: 10.1512/iumj.2008.57.3719.  Google Scholar

[12]

I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 453-469. doi: 10.1088/0951-7715/19/2/012.  Google Scholar

[13]

I. Kukavica and M. Ziane, Navier-Stokes equations with regularity in one direction, J. Math. Phys., 48 (2007), 065203. doi: 10.1063/1.2395919.  Google Scholar

[14]

Y. Zhou and M. Pokorný, On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component, J. Math. Phys., 50 (2009), 123514. doi: 10.1063/1.3268589.  Google Scholar

[15]

Y. Zhou and M. Pokorný, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107. doi: 10.1088/0951-7715/23/5/004.  Google Scholar

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