  Given an initial data $v_0$ with vorticity $\Om_0=\na\times v_0$ in $L^{\frac
3 2},$ (which implies that $v_0$ belongs to the Sobolev space $H^{\frac12}$),
we prove that the solution $v$ given by the classical Fujita-Kato theorem blows
up in a finite time $T^\star$ only if, for any $p$ in $ ]4,6[$ and any unit
vector $e$ in $\R^3,$ there holds $ \int_0^{T^\star}\|v(t)\cdot
e\|_{\dH^{\f12+\f2p}}^p\,dt=\infty.$ We remark that all these quantities are
scaling invariant under the scaling transformation of Navier-Stokes system.
