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Kolmogorov's dissipation number and the number of degrees of freedom for the 3D Navier–Stokes equations

Part of: Equations of mathematical physics and other areas of application Infinite-dimensional dissipative dynamical systems

Published online by Cambridge University Press:  15 January 2019

Alexey Cheskidov  and
Mimi Dai
Alexey Cheskidov
Affiliation:
Department of Mathematics, Stat. and Comp. Sci., University of Illinois Chicago, Chicago, IL 60607, USA ([email protected]; [email protected])
Mimi Dai
Affiliation:
Department of Mathematics, Stat. and Comp. Sci., University of Illinois Chicago, Chicago, IL 60607, USA ([email protected]; [email protected])
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Abstract

Kolmogorov's theory of turbulence predicts that only wavenumbers below some critical value, called Kolmogorov's dissipation number, are essential to describe the evolution of a three-dimensional (3D) fluid flow. A determining wavenumber, first introduced by Foias and Prodi for the 2D Navier–Stokes equations, is a mathematical analogue of Kolmogorov's number. The purpose of this paper is to prove the existence of a time-dependent determining wavenumber for the 3D Navier–Stokes equations whose time average is bounded by Kolmogorov's dissipation wavenumber for all solutions on the global attractor whose intermittency is not extreme.

Keywords

Navier–Stokes equationsdetermining modesglobal attractor

MSC classification

Primary: 35Q35: PDEs in connection with fluid mechanics
Secondary: 37L30: Attractors and their dimensions, Lyapunov exponents
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 2 , April 2019 , pp. 429 - 446
Copyright
Copyright © Royal Society of Edinburgh 2019 

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