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 three-dimensional space with an initial condition from W0,σ1,2, T>0T>0 and u∈Lt(0,T;Ls)u∈Lt(0,T;Ls), where 2/t+3/s=59/302/t+3/s=59/30 for s∈(2,30/13]s∈(2,30/13] and 2/t+3/s=7/4+1/(2s)2/t+3/s=7/4+1/(2s) for s∈(30/13,3)s∈(30/13,3) then u   is regular on (0,T)(0,T). We prove our result as a special case of a more general method which might possibly bring a further improvement.