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 $\{{\tilde{\mathrm{\Omega }}}_m(t)\}$ are used which are based on $L^{2m}$-norms of the vorticity. To avoid using derivatives a closure is assumed that suggests that the ${\tilde{\mathrm{\Omega }}}_m$ ($m>1$) are slaved to ${\tilde{\mathrm{\Omega }}}_1$ (the global enstrophy) in the form $\tilde{\Omega}{m} = \tilde{\Omega}{1}$\mathcal{F}${m}(\tilde{\Omega}{1})$. This is shaped by the constraint of two H"older 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 $1 \leq \lambda_{m}(\tau) \leq 4$,wherecurvesofconstant$\lambda_{m}$ form the boundaries between tongue-shaped regions. In regions where $2.5 \leq \lambda_{m} \leq 4$and$1 \leq \lambda_{m} \leq 2$ the Navier-Stokes equations are shown to be regular,: numerical simulations appear to lie in the latter region. Only in the central region $2<{\lambda }_m<2.5$ has no proof of regularity been found.