Global solutions of $3$-D Navier-Stokes system with small unidirectional derivative

Abstract

Given initial data $u_0=(u_0^\h,u_0^3)\in H^{-\d,0}\cap H^{\f12}(\R^3)$ with both~$\uh_0$ and~$\nabla_{\rm h}\uh_0$ belonging to ~$L^2(\R^3)\cap L^\infty(\R_\v;L^2(\R^2_\h))$ and $u_0^\h\in L^\infty(\R_\v, H^{-\d}(\R^2_\h))$ for some $\delta \in ]0,1[,$ if in addition $\pa_3u_0$ belongs to $H^{-\frac{1}{2},0}\cap H^{\frac{1}{2},0}({\mathbb{R}}^3),$ we prove that the classical $3$-D Navier-Stokes system has a unique global Fujita-Kato solution provided that $\|\pa_3u_0\|_{H^{-\f12,0}}$ is sufficiently small compared to a constant which depends only on the norms of the initial data. In particular, this result provides some classes of large initial data which generate unique global solutions to 3-D Navier-Stokes system.