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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.

Long time decay of 3D-NSE in Lei-Lin-Gevrey spaces