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 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 $1\le {\lambda }_m(\tau )\le 4$, where curves of constant ${\lambda }_m$ form the boundaries between tongue-shaped regions. In regions where $2.5\le {\lambda }_m\le 4$ and $1\le {\lambda }_m\le 2$ 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.

10 pages; 2 figures

References in corpus (3)

• The multifractal nature of turbulent energy dissipation
• Continuous Data Assimilation for the 2D Bénard Convection through Velocity Measurements Alone
• Depletion of Nonlinearity in Magnetohydrodynamic Turbulence: Insights from Analysis and Simulations