Asymptotic behavior for the Navier--Stokes Equations with nonzero external forces
Résumé
We obtain the lower bounds of the temporal-spatial decays for weak solutions of the Navier-Stokes equations $ C_0(1+t)^{-\frac{5-2i}{4}}\le \|(1+|{\cdot}|^2)^{i/2}\bu(\cdot,t)\|_{L^2} \le C_1(1+t)^{-\frac{5-2i}{4}} $ for $i=1,2$. The upper bound parts are estimated in several papers, for example [Bae, Jin, Upper and lower bounds of temporal and spatial decays for the Navier--Stokes equations, J. Diff. Eq. 209, 365--391 (2006)] and [He, Xin, On the decay properties of solutions to the non-stationary Navier-Stokes equations in $R^3$, Proc. Roy. Soc. Edinburgh Sect. A, 131 no. 3 (2001) 597--619. By the interpolation arguments, we also have $C_0(1+t)^{-\frac{5}{4}+\frac{\alpha}{2}} \le \|(1+|x|^2)^{\alpha/2}\mathbf u(\cdot,t)\|_{L^2} \le C_1(1+t)^{-\frac{5}{4}+\frac{\alpha}{2}} $ for $0\le \alpha\le 2$.