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@article{Gibbon:2015:imamat/hxv037,
author = {Gibbon, JD},
doi = {imamat/hxv037},
journal = {IMA Journal of Applied Mathematics},
pages = {308--320},
title = {High-low frequency slaving and regularity issues in the 3D Navier-Stokesequations},
url = {http://dx.doi.org/10.1093/imamat/hxv037},
volume = {81},
year = {2015}
}

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TY  - JOUR
AB  - The old idea that an infinite dimensional dynamical system may have its high modes or frequenciesslaved to low modes or frequencies is re-visited in the context of the 3D Navier-Stokes equations. A setof dimensionless frequencies {˜ m(t)} are used which are based on L2m-norms of the vorticity. To avoidusing derivatives a closure is assumed that suggests that the ˜ m (m > 1) are slaved to ˜1 (the globalenstrophy) in the form ˜ m = ˜1Fm(˜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 numericalNavier-Stokes and MHD simulations. When written as a phase plane in a scaled form, this relation isparametrized by a set of functions 1 6 λm(τ) 6 4, where curves of constant λm form the boundariesbetween tongue-shaped regions. In regions where 2.5 6 λm 6 4 and 1 6 λm 6 2 the Navier-Stokesequations are shown to be regular : numerical simulations appear to lie in the latter region. Only in thecentral region 2 < λm < 2.5 has no proof of regularity been found.
AU  - Gibbon,JD
DO  - imamat/hxv037
EP  - 320
PY  - 2015///
SN  - 0272-4960
SP  - 308
TI  - High-low frequency slaving and regularity issues in the 3D Navier-Stokesequations
T2  - IMA Journal of Applied Mathematics
UR  - http://dx.doi.org/10.1093/imamat/hxv037
UR  - http://hdl.handle.net/10044/1/27524
VL  - 81
ER  -

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