Regularity for 3d Navier-stokes Equations in Terms of Two Components of the Vorticity

We establish regularity conditions for the 3D Navier-Stokes equation via two components of the vorticity vector. It is known that if a LerayHopf weak solution u satisfies ω̃ ∈ L2/(2−r)(0, T ;L(R)) with 0 < r < 1, where ω̃ form the two components of the vorticity, ω = curlu, then u becomes the classical solution on (0, T ] (see [5]). We prove the regularity of Leray-Hopf weak solution u under each of the following two (weaker) conditions: ω̃ ∈ L2/(2−r)(0, T ;Ṁ2,3/r(R)) for 0 < r < 1, ∇ũ ∈ L2/(2−r)(0, T ;Ṁ2,3/r(R)) for 0 ≤ r < 1, where Ṁ2,3/r(R) is the Morrey-Campanato space. Since L3/r(R3) is a proper subspace of Ṁ2,3/r(R), our regularity criterion improves the results in ChaeChoe [5].