Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces

In this paper, we study the long time decay of global solution to the 3D incompressible Navier-Stokes equations. We prove that if u ∈ C ( R + , X − 1 , σ ( R 3 ) ) u\in {\mathcal{C}}\left({{\mathbb{R}}}^{+},{{\mathcal{X}}}^{-1,\sigma }\left({{\mathbb{R}}}^{3})) is a global solution to the considered equation, where X i , σ ( R 3 ) {{\mathcal{X}}}^{i,\sigma }\left({{\mathbb{R}}}^{3}) is the Fourier-Lei-Lin space with parameters i = − 1 i=-1 and σ ≥ − 1 \sigma \ge -1 , then ‖ u ( t ) ‖ X − 1 , σ \Vert u\left(t){\Vert }_{{{\mathcal{X}}}^{-1,\sigma }} decays to zero as time goes to infinity. The used techniques are based on Fourier analysis.