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 equationsC0(1+t)-5-2i4⩽∥(1+|·|2)i/2u(·,t)∥L2⩽C1(1+t)-5-2i4for 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 haveC0(1+t)-54+α2⩽∥(1+|x|2)α/2u(·,t)∥L2⩽C1(1+t)-54+α2for 0⩽α⩽2.