  In this paper we construct two families of initial data being arbitrarily
large under any scaling-invariant norm for which their corresponding weak
solution to the three-dimensional Navier-Stokes equations become smooth on
either $[0,T_1]$ or $ [T_2,\infty)$, respectively, where $T_1$ and $T_2$ are
two times prescribed previously. In particular, $T_1$ can be arbitrarily large
and $T_2$ can be arbitrarily small. Therefore, possible formation of
singularities would occur after a very long or short evolution time,
respectively.
  We further prove that if a large part of the kinetic energy is consumed prior
to the first (possible) blow-up time, then the global-in-time smoothness of the
solutions follows for the two families of initial data.
