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

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.