American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

Daoyuan Fang, Chenyin Qian. Regularity criterion for 3D Navier-Stokes equations in Besov spaces. Communications on Pure & Applied Analysis, 2014, 13 (2) : 585-603. doi: 10.3934/cpaa.2014.13.585
[2]

Zujin Zhang. A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component. Communications on Pure & Applied Analysis, 2013, 12 (1) : 117-124. doi: 10.3934/cpaa.2013.12.117
[3]

Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747
[4]

Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319
[5]

Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717
[6]

Patrick Penel, Milan Pokorný. Improvement of some anisotropic regularity criteria for the Navier--Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1401-1407. doi: 10.3934/dcdss.2013.6.1401
[7]

Jishan Fan, Yasuhide Fukumoto, Yong Zhou. Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations. Kinetic & Related Models, 2013, 6 (3) : 545-556. doi: 10.3934/krm.2013.6.545
[8]

Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141
[9]

Hongjie Dong, Kunrui Wang. Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5289-5323. doi: 10.3934/dcds.2020228
[10]

Zijin Li, Xinghong Pan. Some Remarks on regularity criteria of Axially symmetric Navier-Stokes equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1333-1350. doi: 10.3934/cpaa.2019064
[11]

Keyan Wang. On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1067-1072. doi: 10.3934/cpaa.2009.8.1067
[12]

Hui Chen, Daoyuan Fang, Ting Zhang. Regularity of 3D axisymmetric Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1923-1939. doi: 10.3934/dcds.2017081
[13]

Yukang Chen, Changhua Wei. Partial regularity of solutions to the fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5309-5322. doi: 10.3934/dcds.2016033
[14]

Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209
[15]

Alessio Falocchi, Filippo Gazzola. Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021151
[16]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[17]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
[18]

Mimi Dai, Han Liu. Low modes regularity criterion for a chemotaxis-Navier-Stokes system. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2713-2735. doi: 10.3934/cpaa.2020118
[19]

Peter Constantin, Gregory Seregin. Global regularity of solutions of coupled Navier-Stokes equations and nonlinear Fokker Planck equations. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1185-1196. doi: 10.3934/dcds.2010.26.1185
[20]

Ahmad Mohammad Alghamdi, Sadek Gala, Chenyin Qian, Maria Alessandra Ragusa. The anisotropic integrability logarithmic regularity criterion for the 3D MHD equations. Electronic Research Archive, 2020, 28 (1) : 183-193. doi: 10.3934/era.2020012

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (117)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy