  In this paper, we prove that there exists a unique global solution of $3D$
Navier-Stokes equation if
$\exp(a|D|^{1/\sigma})u^0\in{\mathcal{X}}^{-1}(\mathbb R^3)$ and
$\|u^0\|_{{\mathcal{X}}^{-1}}<\nu$. Moreover, we will show that
$\|\exp(a|D|^{1/\sigma}) u(t)\|_{{\mathcal{X}}^{-1}}$ goes to zero if the time
$t$ goes to infinity.
