We study some dyadic models for incompressible magnetohydrodynamics and Navier–Stokes equation. The existence of fixed point and stability of the fixed point are established. The scaling law of Kolmogorov’s dissipation wavenumber arises from heuristic analysis. In addition, a time-dependent determining wavenumber is shown to exist; moreover, the time average of the determining wavenumber is proved to be bounded above by Kolmogorov’s dissipation wavenumber. Additionally, based on the knowledge of the fixed point and stability of the fixed point, numerical simulations are performed to illustrate the energy spectrum in the inertial range below Kolmogorov’s dissipation wavenumber.
Kolmogorov's dissipation number and the number of degrees of freedom for the 3D Navier–Stokes equations
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.
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- Award ID(s):
- 1815069
- NSF-PAR ID:
- 10105416
- Date Published:
- Journal Name:
- Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- Volume:
- 149
- Issue:
- 2
- ISSN:
- 0308-2105
- Page Range / eLocation ID:
- 429 to 446
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/prm.2018.33
