The BKM criterion for the 3D Navier–Stokes equations via two velocity components

In the study of the regularity criterion of Leray-Hopf weak solutions to the 3D Navier–Stokes equations, the Beale–Kato–Majda type criterion is obtained in terms of the horizontal derivatives of the two velocity components ∫0T‖∇hũ(s)‖Ḃ∞,∞0ds<∞,ũ=(u1,u2,0),∇hũ=(∂1ũ,∂2ũ,0).