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High-low frequency slaving and regularity issues in the 3D Navier-Stokes equations

The old idea that an infinite dimensional dynamical system may have its high modes or frequencies slaved to low modes or frequencies is re-visited in the context of the $3D$ Navier-Stokes equations. A set of dimensionless frequencies $\{\tildeΩ_{m}(t)\}$ are used which are based on $L^{2m}$-norms of the vorticity. To avoid using derivatives a closure is assumed that suggests that the $\tildeΩ_{m}$ ($m>1$) are slaved to $\tildeΩ_{1}$ (the global enstrophy) in the form $\tildeΩ_{m} = \tildeΩ_{1}\mathcal{F}_{m}(\tildeΩ_{1})$. This is shaped by the constraint of two Hölder inequalities and a time average from which emerges a form for $\mathcal{F}_{m}$ which has been observed in previous numerical Navier-Stokes and MHD simulations. When written as a phase plane in a scaled form, this relation is parametrized by a set of functions $1 \leq λ_{m}(τ) \leq 4$, where curves of constant $λ_{m}$ form the boundaries between tongue-shaped regions. In regions where $2.5 \leq λ_{m} \leq 4$ and $1 \leq λ_{m} \leq 2$ the Navier-Stokes equations are shown to be regular\,: numerical simulations appear to lie in the latter region. Only in the central region $2 < λ_{m} < 2.5$ has no proof of regularity been found.