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

Abstract

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 $\{\tilde{\Omega}_{m}(t)\}$ are used which are based on $L^{2m}$-norms of the vorticity. To avoid using derivatives a closure is assumed that suggests that the $\tilde{\Omega}_{m}$ ($m>1$) are slaved to $\tilde{\Omega}_{1}$ (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 Holder 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 $1 \leq \lambda_{m}(\tau) \leq 4$, where curves of constant $\lambda_{m}$ 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.