Upper and lower bounds of temporal and spatial decays for the Navier–Stokes equations

We obtain the lower bounds of the temporal–spatial decays for weak solutions of the Navier–Stokes equations for i=1,2. The upper bound parts are estimated in several papers, for example (Bae and Jin (submitted) and He and Xin (Proc. Roy. Soc. Edinburg Section A 131(3) (2001) 597). By the interpolation arguments, we also have for 0 α 2.