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

Abstract

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 $\Vert u^0{\Vert }_{{\mathcal{X}}^{-1}}<\nu$. Moreover, we will show that $\Vert \exp (a|D{|}^{1/\sigma })u(t){\Vert }_{{\mathcal{X}}^{-1}}$ goes to zero if the time $t$ goes to infinity.

15 pages