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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

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 & 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.   Google Scholar

[2]

C. Cao, Sufficient conditions for the regularity to the 3D Navier-Stokes equations,, Discrete Contin. Dyn. Syst., 26 (2010), 1141.  doi: 10.3934/dcds.2010.26.1141.  Google Scholar

[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.  doi: 10.1007/s00205-011-0439-6.  Google Scholar

[4]

L. Escauriaza, G. Seregin and V. Šverák, Backward uniqueness for parabolic equations,, Arch. Ration. Mech. Anal., 169 (2003), 147.   Google Scholar

[5]

I. Kukavica and M. Ziane, Navier-Stokes equations with regularity in one direction,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2395919.  Google Scholar

[6]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace,, (French), 63 (1934), 193.  doi: 10.1007/BF02547354.  Google Scholar

[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, (1999), 391.   Google Scholar

[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.  doi: 10.1023/B:APOM.0000048124.64244.7e.  Google Scholar

[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.  doi: 10.1007/s00021-010-0038-6.  Google Scholar

[10]

G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes,, Ann. Math. Pura Appl. (4), 48 (1959), 173.   Google Scholar

[11]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, Arch. Ration. Mech. Anal., 9 (1962), 187.   Google Scholar

[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).  doi: 10.1063/1.3589966.  Google Scholar

[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).  doi: 10.1063/1.3268589.  Google Scholar

[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.  doi: 10.1088/0951-7715/23/5/004.  Google Scholar

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.   Google Scholar

[2]

C. Cao, Sufficient conditions for the regularity to the 3D Navier-Stokes equations,, Discrete Contin. Dyn. Syst., 26 (2010), 1141.  doi: 10.3934/dcds.2010.26.1141.  Google Scholar

[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.  doi: 10.1007/s00205-011-0439-6.  Google Scholar

[4]

L. Escauriaza, G. Seregin and V. Šverák, Backward uniqueness for parabolic equations,, Arch. Ration. Mech. Anal., 169 (2003), 147.   Google Scholar

[5]

I. Kukavica and M. Ziane, Navier-Stokes equations with regularity in one direction,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2395919.  Google Scholar

[6]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace,, (French), 63 (1934), 193.  doi: 10.1007/BF02547354.  Google Scholar

[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, (1999), 391.   Google Scholar

[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.  doi: 10.1023/B:APOM.0000048124.64244.7e.  Google Scholar

[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.  doi: 10.1007/s00021-010-0038-6.  Google Scholar

[10]

G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes,, Ann. Math. Pura Appl. (4), 48 (1959), 173.   Google Scholar

[11]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, Arch. Ration. Mech. Anal., 9 (1962), 187.   Google Scholar

[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).  doi: 10.1063/1.3589966.  Google Scholar

[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).  doi: 10.1063/1.3268589.  Google Scholar

[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.  doi: 10.1088/0951-7715/23/5/004.  Google Scholar

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