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

TL;DR

This paper investigates the relationship between high and low frequency modes in the 3D Navier-Stokes equations, proposing a slaving hypothesis and analyzing the conditions for regularity based on frequency ratios.

Contribution

It introduces a new closure assumption linking high modes to low modes using dimensionless frequencies and analyzes the regularity of solutions through a phase plane approach.

Findings

Numerical simulations mostly lie in the regular region with 1 ≤ λ_m ≤ 2.

The phase plane analysis identifies boundary regions for regularity.

No proof of regularity exists in the central region 2 < λ_m < 2.5.

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 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 $1 \leq \lambda_{m}(\tau) \leq 4$, 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.