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, a 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.
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