A regularity criterion for the Navier-Stokes equations based on the gradient of one velocity component

We show that if u is a Leray solution to the Navier-Stokes equations in the full threedimensional space with an initial condition from W-o,sigma(1,2), T > 0 and u is an element of L-t(0, T; L-s), where 2/t 3/s = 59/30 for s E (2,30/13] and 2/t 3/s = 7/4 1/(2s) for s is an element of (30/13,3) then u is regular on (0, T). We prove our result as a special case of a more general method which might possibly bring a further improvement. (C) 2016 Elsevier Inc. All rights reserved.