Remarks on regularity criteria for the Navier-Stokes equations via one velocity component

Recently, Pokorny and Zhou proved that if parallel to u(3)parallel to(L infinity (0,T;L) (10/3) ((R3))) << 1 or parallel to del u(3)parallel to(L infinity(0,T;L) (30/19) ((R3))) << 1, then the weak solution u of the 3D Navier-Stokes equations is regular on R-3 x (0, T]. In this paper we remove the smallness assumptions by using a new approach which may be of independent interest and further application. (C) 2013 Elsevier Ltd. All rights reserved.