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An anisotropic regularity criterion for the 3D Navier-Stokes equations
1. | Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, P. R. |
2. | Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China |
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 $\mathbfR^n$, 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 $\mathbfR^n$, 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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