Large time behaviour of solutions to the 3D-NSE in ${\mathcal{X}}^{\sigma }$ spaces

In this paper we study the incompressible Navier-Stokes equations in $L^2({\mathbb{R}}^3)\cap {\mathcal{X}}^{-1}({\mathbb{R}}^3)$. In the global existence case, we establish that if the solution $u$ is in the space $C(\mathbb R^+,L^2\cap\mathcal X^{-1})$,thenfor$\sigma>-3/2$thedecayof$|u(t)|_{\mathcal X^\sigma}$isatleastoftheorderof$t^{-\frac{\sigma+\frac{3}{2}}{2}}$. Fourier analysis and standard techniques are used.