Improvement of some anisotropic regularity criteria for the Navier--Stokes equations

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October 2013, 6(5): 1401-1407. doi: 10.3934/dcdss.2013.6.1401

Improvement of some anisotropic regularity criteria for the Navier--Stokes equations

Patrick Penel ^1, and Milan Pokorný ^2,

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

Received November 2011 Revised February 2012 Published March 2013

Abstract
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We consider the incompressible Navier--Stokes equations in the entire three-dimensional space. Assuming additional regularity on the components of the vector field $\partial_3$u we show intermediate anisotropic regularity results between the results by Kukavica and Ziane [5] and by Cao and Titi [3]and improve the result from the paper by Penel and Pokorný [9].

Keywords: regularity of solution, Navier-Stokes equations, regularity criteria..

Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D0.

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

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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