On the regularity criterion for the Navier–Stokes equations involving the diagonal entry of the velocity gradient

In this paper, we provide a regularity criterion for the Navier–Stokes equations involving the diagonal entry of the velocity gradient, which says that if ∂3u3∈L∞(0,T;L2(R3)),∂3u3∈L∞(0,T;L2(R3)), then the weak solution is smooth on (0,T)(0,T). This is an important case in the sense that the strong solution is in this class. Moreover, we verify the limiting case of one regularity criterion in Cao and Titi (2011).