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

Recently, Pokorný and Zhou proved that if ‖u3‖L∞(0,T;L103(R3))≪1 or ‖∇u3‖L∞(0,T;L3019(R3))≪1, then the weak solution uu of the 3D Navier–Stokes equations is regular on R3×(0,T]R3×(0,T]. In this paper we remove the smallness assumptions by using a new approach which may be of independent interest and further application.