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

Abstract

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(R3))u\in {\mathcal{C}}\left({{\mathbb{R}}}^{+},{{\mathcal{X}}}^{-1,\sigma }\left({{\mathbb{R}}}^{3})) is a global solution to the considered equation, where Xi,σ(R3){{\mathcal{X}}}^{i,\sigma }\left({{\mathbb{R}}}^{3}) is the Fourier-Lei-Lin space with parameters i=−1i=-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.

Keywords

• navier-stokes equations
• critical spaces
• long time decay
• 35q30
• 35d35