Abstract
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 is an element of [2,3], we get that the solution is regular if del u(3) is an element of L-t(0, T; L-s(R-3)), 2/t+ 3/s <= 23/12. (C) 2009 American Institute of Physics. [doi: 10.1063/1.3268589]