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A note on local interior regularity of a suitable weak solution to the Navier--Stokes problem
Improvement of some anisotropic regularity criteria for the Navier--Stokes equations
1. | Mathématique et Laboratoire SNC, Université du Sud, Toulon-Var, BP 20132, 83957 La Garde Cedex |
2. | Mathematical Institute of Charles University, Sokolovská 83, 186 75 Praha 8 |
References:
[1] |
H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in $R^n$, Chin. Ann. Math. Ser. B, 16 (1995), 407-412. |
[2] |
C. Cao, Sufficient conditions for the regularity to the 3D Navier-Stokes equations, Discrete Contin. Dyn. Syst., 26 (2010), 1141-1151.
doi: 10.3934/dcds.2010.26.1141. |
[3] |
C. Cao and E. S. Titi, Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor, Arch. Ration. Mech. Anal., 202 (2011), 919-932.
doi: 10.1007/s00205-011-0439-6. |
[4] |
L. Escauriaza, G. Seregin and V. Šverák, Backward uniqueness for parabolic equations, Arch. Ration. Mech. Anal., 169 (2003), 147-157. |
[5] |
I. Kukavica and M. Ziane, Navier-Stokes equations with regularity in one direction, J. Math. Phys., 48 (2007), 065203, 10 pp.
doi: 10.1063/1.2395919. |
[6] |
J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, (French), Acta Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
[7] |
J. Neustupa and P. Penel, Regularity of a suitable weak solution to the Navier-Stokes equations as a consequence of regularity of one velocity component, in "Applied Nonlinear Analysis," Kluwer/Plenum, New York, (1999), 391-402. |
[8] |
P. Penel and M. Pokorný, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity, Appl. Math., 49 (2004), 483-493.
doi: 10.1023/B:APOM.0000048124.64244.7e. |
[9] |
P. Penel and M. Pokorný, On anisotropic regularity criteria for the solutions to 3D Navier-Stokes equations, J. Math. Fluid Mech., 13 (2011), 341-353.
doi: 10.1007/s00021-010-0038-6. |
[10] |
G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Math. Pura Appl. (4), 48 (1959), 173-182. |
[11] |
J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 9 (1962), 187-195. |
[12] |
Z. Zhang, Z. Yao, M. Lu and L. Ni, Some Serrin-type regularity criteria for weak solutions to the Navier-Stokes equations, J. Math. Phys., 52 (2011), 053103, 7 pp.
doi: 10.1063/1.3589966. |
[13] |
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, 11 pp.
doi: 10.1063/1.3268589. |
[14] |
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] |
H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in $R^n$, Chin. Ann. Math. Ser. B, 16 (1995), 407-412. |
[2] |
C. Cao, Sufficient conditions for the regularity to the 3D Navier-Stokes equations, Discrete Contin. Dyn. Syst., 26 (2010), 1141-1151.
doi: 10.3934/dcds.2010.26.1141. |
[3] |
C. Cao and E. S. Titi, Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor, Arch. Ration. Mech. Anal., 202 (2011), 919-932.
doi: 10.1007/s00205-011-0439-6. |
[4] |
L. Escauriaza, G. Seregin and V. Šverák, Backward uniqueness for parabolic equations, Arch. Ration. Mech. Anal., 169 (2003), 147-157. |
[5] |
I. Kukavica and M. Ziane, Navier-Stokes equations with regularity in one direction, J. Math. Phys., 48 (2007), 065203, 10 pp.
doi: 10.1063/1.2395919. |
[6] |
J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, (French), Acta Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
[7] |
J. Neustupa and P. Penel, Regularity of a suitable weak solution to the Navier-Stokes equations as a consequence of regularity of one velocity component, in "Applied Nonlinear Analysis," Kluwer/Plenum, New York, (1999), 391-402. |
[8] |
P. Penel and M. Pokorný, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity, Appl. Math., 49 (2004), 483-493.
doi: 10.1023/B:APOM.0000048124.64244.7e. |
[9] |
P. Penel and M. Pokorný, On anisotropic regularity criteria for the solutions to 3D Navier-Stokes equations, J. Math. Fluid Mech., 13 (2011), 341-353.
doi: 10.1007/s00021-010-0038-6. |
[10] |
G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Math. Pura Appl. (4), 48 (1959), 173-182. |
[11] |
J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 9 (1962), 187-195. |
[12] |
Z. Zhang, Z. Yao, M. Lu and L. Ni, Some Serrin-type regularity criteria for weak solutions to the Navier-Stokes equations, J. Math. Phys., 52 (2011), 053103, 7 pp.
doi: 10.1063/1.3589966. |
[13] |
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, 11 pp.
doi: 10.1063/1.3268589. |
[14] |
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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