High-low frequency slaving and regularity issues in the 3D Navier-Stokes
equations
equations
File(s)
Accepted version
Author(s)
Gibbon, JD
Type
Journal Article
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 {Ω˜ 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 Ω˜ m (m > 1) are slaved to Ω˜
1 (the global
enstrophy) in the form Ω˜ m = Ω˜
1Fm(Ω˜
1). This is shaped by the constraint of two Holder 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 6 λm(τ) 6 4, where curves of constant λm form the boundaries
between tongue-shaped regions. In regions where 2.5 6 λm 6 4 and 1 6 λm 6 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.
slaved to low modes or frequencies is re-visited in the context of the 3D Navier-Stokes equations. A set
of dimensionless frequencies {Ω˜ 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 Ω˜ m (m > 1) are slaved to Ω˜
1 (the global
enstrophy) in the form Ω˜ m = Ω˜
1Fm(Ω˜
1). This is shaped by the constraint of two Holder 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 6 λm(τ) 6 4, where curves of constant λm form the boundaries
between tongue-shaped regions. In regions where 2.5 6 λm 6 4 and 1 6 λm 6 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.
Date Issued
2015-11-24
Date Acceptance
2015-11-01
Citation
IMA Journal of Applied Mathematics, 2015, 81 (2), pp.308-320
ISSN
0272-4960
Publisher
Oxford University Press (OUP)
Start Page
308
End Page
320
Journal / Book Title
IMA Journal of Applied Mathematics
Volume
81
Issue
2
Copyright Statement
This is a pre-copyedited, author-produced PDF of an article accepted for publication in IMA Journal of Applied Mathematics following peer review. The version of record J. D. Gibbon
High–low frequency slaving and regularity issues in the 3D Navier–Stokes equations
IMA J Appl Math (2016) 81 (2): 308-320 first published online November 24, 2015 is available online at: https://dx.doi.org/10.1093/imamat/hxv037
High–low frequency slaving and regularity issues in the 3D Navier–Stokes equations
IMA J Appl Math (2016) 81 (2): 308-320 first published online November 24, 2015 is available online at: https://dx.doi.org/10.1093/imamat/hxv037
Subjects
Science & Technology
Physical Sciences
Mathematics, Applied
Mathematics
3D Navier-Stokes equations
closure
existence
INERTIAL MANIFOLDS
NUMERICAL SIMULATIONS
DETERMINING MODES
TURBULENCE
DISSIPATION
ENSTROPHY
MOMENTS
NUMBER
BOUNDS
FLUID
Applied Mathematics
0102 Applied Mathematics
Publication Status
Published