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

作者：Gibbon J D^*

来源：IMA Journal of Applied Mathematics, 2016, 81(2): 308-320.

DOI：10.1093/imamat/hxv037

• 3D Navier-Stokes equations
• closure
• existence

The old idea that an infinite-dimensional dynamical system may have its high modes or frequencies slaved to low modes or frequencies is revisited in the context of the 3D Navier-Stokes equations. A set of dimensionless frequencies {(Omega) over tilde (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 (Omega) over tilde (m) (m > 1) are slaved to (Omega) over tilde (1) (the global enstrophy) in the form (Omega) over tilde (m) = (Omega) over tilde (1) F-m((Omega) over tilde (1)). This is shaped by the constraint of two Hlder inequalities and a time average from which emerges a form for F-m which has been observed in previous numerical Navier-Stokes and magneto-hydrodynamic simulations. When written as a phase plane in a scaled form, this relation is parametrized by a set of functions 1 <= lambda(m)(tau) <= 4, where curves of constant.m form the boundaries between tongue-shaped regions. In regions where 2.5 <= lambda(m) <= 4 and 1 <= lambda(m) <= 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.

• 出版日期2016-4