Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in R3

We consider the regularity of weak solutions to the Navier-Stokes equations in R3. Let u be a weak solution in R3×(0,T), w=curlu, and w˜=(w1,w2,0). It is proved that u becomes a classical solution if w˜∈Lq(0,T;B˙r,σ0), for 2/q+3/r=2,32