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

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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 {Ωₘ(t)} are used which are based on L²ᵐ-norms of the vorticity. To avoid using derivatives a closure is assumed that suggests that the Ωₘ (m>1) are slaved to Ω₁ (the global enstrophy) in the form Ωₘ = Ω₁Fₘ(Ω₁). This is shaped by the constraint of two Hölder inequalities and a time average from which emerges a form for Fₘ 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 ≤ λₘ(τ) ≤ 4, where curves of constant λₘ form the boundaries between tongue-shaped regions. In regions where 2.5 ≤ λₘ ≤ 4 and 1 ≤ λₘ ≤ 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 < λₘ < 2.5 has no proof of regularity been found.