  Higher moments of the vorticity field $\Omega_{m}(t)$ in the form of
$L^{2m}$-norms ($1 \leq m < \infty$) are used to explore the regularity problem
for solutions of the three-dimensional incompressible Navier-Stokes equations
on the domain $[0, L]^{3}_{per}$. It is found that the set of quantities $$
D_{m}(t) = \Omega_{m}^{\alpha_{m}} ,\qquad\qquad\alpha_{m} = \frac{2m}{4m-3},
$$ provide a natural scaling in the problem resulting in a bounded set of time
averages $<D_{m}>_{T}$ on a finite interval of time $[0, T]$. The behaviour of
$D_{m+1}/D_{m}$ is studied on what are called `good' and `bad' intervals of
$[0, T]$ which are interspersed with junction points (neutral) $\tau_{i}$. For
large but finite values of $m$ with large initial data \big($\Omega_{m}(0) \leq
\varpi_{0}O(\Gr^{4})$\big), it is found that there is an upper bound $$
\Omega_{m} \leq c_{av}^{2}\varpi_{0}\Gr^{4} ,\qquad\varpi_{0} = \nu L^{-2}, $$
which is punctured by infinitesimal gaps or windows in the vertical walls
between the good/bad intervals through which solutions may escape. While this
result is consistent with that of Leray \cite{Leray} and Scheffer
\cite{Scheff76}, this estimate for $\Omega_{m}$ corresponds to a length scale
well below the validity of the Navier-Stokes equations.
