On the critical one component regularity for 3-D Navier-Stokes system

Abstract

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 $H^{\frac{1}{2}}$), 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 ${\mathbb{R}}^3,$ 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.