On a regularity criterion for the Navier–Stokes equations involving gradient of one velocity component

We improve the regularity criterion for the incompressible Navier–Stokes equations in the full three-dimensional space involving the gradient of one velocity component. The method is based on recent results of Cao and Titi [see “Regularity criteria for the three dimensional Navier–Stokes equations,” Indiana Univ. Math. J. 57, 2643 (2008)] and Kukavica and Ziane [see “Navier-Stokes equations with regularity in one direction,” J. Math. Phys. 48, 065203 (2007)]. In particular, for s∊[2,3], we get that the solution is regular if ∇u3∊Lt(0,T;Ls(R3)), 2/t+3/s≤2312.

[1]  Yong Zhou,et al.  On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in $$\mathbb{R}^{N}$$ , 2006 .

[2]  Patrick Penel,et al.  Anisotropic and Geometric Criteria for Interior Regularity of Weak Solutions to the 3D Navier—Stokes Equations , 2001 .

[3]  Milan Pokorny.  On the result of He concerning the smoothness of solutions to the Navier-Stokes equations , 2003 .

[4]  Yong Zhou.  Weighted regularity criteria for the three-dimensional Navier—Stokes equations , 2009, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[5]  Yong Zhou,et al.  Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain , 2004 .

[6]  Edriss S. Titi,et al.  Regularity Criteria for the Three-dimensional Navier-Stokes Equations , 2008 .

[7]  Yong Zhou,et al.  A new regularity criterion for weak solutions to the Navier–Stokes equations , 2005 .

[8]  Igor Kukavica,et al.  One component regularity for the Navier–Stokes equations , 2006 .

[9]  Milan Pokorný,et al.  Some New Regularity Criteria for the Navier-Stokes Equations Containing Gradient of the Velocity , 2004 .

[10]  Yong Zhou,et al.  On the regularity of the solutions of the Navier–Stokes equations via one velocity component , 2010 .

[11]  Yong Zhou,et al.  A New Regularity Criterion for the Navier-Stokes Equations in Terms of the Direction of Vorticity , 2005 .

[12]  V. Sverák,et al.  Backward Uniqueness for Parabolic Equations , 2003 .

[13]  Yong Zhou,et al.  A New Regularity Criterion for the Navier-Stokes Equations in Terms of the Gradient of One Velocity Component , 2002 .

[14]  Luigi C. Berselli,et al.  On the regularizing effect of the vorticity direction in incompressible viscous flows , 2002, Differential and Integral Equations.

[15]  Song Jiang,et al.  On regularity criteria for the n-dimensional Navier–Stokes equations in terms of the pressure , 2008 .

[16]  Igor Kukavica,et al.  Navier-Stokes equations with regularity in one direction , 2007 .

[17]  Jean Leray,et al.  Sur le mouvement d'un liquide visqueux emplissant l'espace , 1934 .

[18]  A short note on regularity criteria for the Navier–Stokes equations containing the velocity gradient , 2005 .

[19]  G. Prodi.  Un teorema di unicità per le equazioni di Navier-Stokes , 1959 .