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

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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

Abstract
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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 and 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.
[2]	E. Hopf, Über die Anfangwertaufgaben für die hydromischen Grundgleichungen, Math. Nachr., 4 (1951), 213-321. doi: 10.1002/mana.3210040121.
[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.
[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.
[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.
[6]	H. Beirão da Veiga, A new regularity class for the Navier-stokes equations in $\mathbf{R}^n$, Chin. Ann. Math., 16 (1995), 407-412.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[2]	E. Hopf, Über die Anfangwertaufgaben für die hydromischen Grundgleichungen, Math. Nachr., 4 (1951), 213-321. doi: 10.1002/mana.3210040121.
[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.
[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.
[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.
[6]	H. Beirão da Veiga, A new regularity class for the Navier-stokes equations in $\mathbf{R}^n$, Chin. Ann. Math., 16 (1995), 407-412.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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