AN ANISOTROPIC REGULARITY CRITERION FOR THE 3D NAVIER-STOKES EQUATIONS

In this paper, we establish an anisotropic regularity criterion for the 3D incompressible Navier-Stokes equations. It is proved that a weak solution u is regular on [0, T], provided partial derivative u(3)/partial derivative x(3) is an element of L-t1 (0, T; L-s1 (R-3)), with 2/t(1)+3/s(1) <= 2, S-1 is an element of (3/2, +infinity] and del(h)u(3) is an element of L-t2 (0, T; L-s2 (R-3)), with either 2/t(2)+3/s(2) <= 19/12 + 1/2s(2), s(2) is an element of (30/19, 3] or 2/t(2) + 3/s(2) <= 3/2 + 3/4s(2), s(2) is an element of (3, +infinity]. Our result in fact improves a regularity criterion of Zhou and Pokorny [Nonlinearity 23 (2010), 1097-11071].