Long time decay for 3D Navier-Stokes equations in Sobolev-Gevrey spaces

In this article, we study the long time decay of global solution to 3D incompressible Navier-Stokes equations. We prove that if $u\in{\mathcal C}([0,\infty),H^1_{a,\sigma}(\mathbb{R}^3))$ is a global solution, where $H^1_{a,\sigma}(\mathbb{R}^3)$ is the Sobolev-Gevrey spaces with parameters $a>0$ and $\sigma>1$, then $\|u(t)\|_{H^1_{a,\sigma}(\mathbb{R}^3)}$ decays to zero as time approaches infinity. Our technique is based on Fourier analysis.