Regularity criteria for the Navier-Stokes equations based on one component of velocity

We study the regularity criteria for the incompressible Navier-Stokes equations in the whole space R-3 based on one velocity component, namely u(3), del u(3) and del(2)u(3). We use a generalization of the Troisi inequality and anisotropic Lebesgue spaces and prove, for example, that the condition del u(3) is an element of L-beta(0, T; L-P), where 2/beta + 3/p = 7/4 + 1/(2p) and p is an element of (2, infinity], yields the regularity of u on (0, T]. (C) 2016 Elsevier Ltd. All rights reserved.