American Institute of Mathematical Sciences

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

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

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].
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

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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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