Long time decay for 3D-NSE in Gevrey-Sobolev spaces

In this paper we prove, if $u$ is a global solution to Navier-Stokes equations in the Sobolev-Gevrey spaces $H_{a,\sigma }^1({\mathbb{R}}^3)$, then $\mathrm{\parallel }u(t){\mathrm{\parallel }}_{H_{a,\sigma }^1}$ decays to zero as time goes to infinity. Fourier analysis is used.