We consider the regularity criteria for a weak solution $\mathit{u}=(u_1,u_2,u_3)$ to the Navier-Stokes equations in ${\mathbb{R}}^3$ . Denoting by $\mathit{\omega }=({\omega }_1,{\omega }_2,{\omega }_3)=\text{curl}\mathit{u}$ the vorticity, we then prove that $\mathit{u}$ is smooth, provided u₃, ω₃; or ∂₃u₃, ω₃ are in some Serrin-type integrability classes.

We study the Cauchy problem for the 3D Navier-Stokes equations, and prove some scalaring-invariant regularity criteria involving only one velocity component.