A remark on the regularity criterion for the 3D Navier–Stokes equations involving the gradient of one velocity component

In this paper, we consider the regularity criterion for weak solutions to the 3D Navier–Stokes equations. We show that if ∇u3∇u3 belongs to some multiplier spaces, then the solution actually is smooth on (0,T)(0,T). In particular, we have the regularity condition in the BMO   spaces: ∇u3∈L43(0,T;BMO), which improves previous results.