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 , .