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Regularity and singularity in solutions of the three-dimensional Navier-Stokes equations

Higher moments of the vorticity field $Ω_{m}(t)$ 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 $$ D_{m}(t) = Ω_{m}^{α_{m}} ,\qquad\qquadα_{m} = \frac{2m}{4m-3}, $$ 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) $τ_{i}$. For large but finite values of $m$ with large initial data \big($Ω_{m}(0) \leq \varpi_{0}O(\Gr^{4})$\big), it is found that there is an upper bound $$ Ω_{m} \leq c_{av}^{2}\varpi_{0}\Gr^{4} ,\qquad\varpi_{0} = νL^{-2}, $$ 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 $Ω_{m}$ corresponds to a length scale well below the validity of the Navier-Stokes equations.