A regularity criterion for the solutions of 3D Navier–Stokes equations

In terms of two partial derivatives of any two components of velocity fields, we give a new criterion for the regularity of solutions of the Navier–Stokes equation in R3. More precisely, let u = (u1, u2, u3) be a weak solution in (0, T )×R3. Then u becomes a classical solution if any two functions of ∂1u1, ∂2u2 and ∂3u3 belong to Lθ (0, T ; Lr (R3)) provided with 2 θ + 3r = 2, 32