Regularity criterion via two components of vorticity on weak solutions to the Navier–Stokes equations in

We consider the regularity of weak solutions to the Navier–Stokes equations in . Let u be a weak solution in , , and . It is proved that u becomes a classical solution if , for , and σ 2r/3. This is an improvement of the result given by Kozono–Yatsu [Extension criterion via two-components of vorticity on strong solution to the 3D Navier–Stokes equations, Math. Z. 246 (2003) 55–68].