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Regimes of nonlinear depletion and regularity in the 3D Navier–Stokes equations
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Gibbon, John D., Donzis, Diego A., Gupta, Anupam, Kerr, Robert M. (Robert McDougall), Pandit, Rahul and Vincenzi, Dario (2014) Regimes of nonlinear depletion and regularity in the 3D Navier–Stokes equations. Nonlinearity, Volume 27 (Number 10). pp. 2605-2625. doi:10.1088/0951-7715/27/10/2605 ISSN 0951-7715.
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Official URL: http://dx.doi.org/10.1088/0951-7715/27/10/2605
Abstract
The periodic 3D Navier–Stokes equations are analyzed in terms of dimensionless, scaled, L2m-norms of vorticity Dm (1 ≤ m < ∞). The first in this hierarchy, D1, is the global enstrophy. Three regimes naturally occur in the D1 − Dm plane. Solutions in the first regime, which lie between two concave curves, are shown to be regular, owing to strong nonlinear depletion. Moreover, numerical experiments have suggested, so far, that all dynamics lie in this heavily depleted regime [1]; new numerical evidence for this is presented. Estimates for the dimension of a global attractor and a corresponding inertial range are given for this regime. However, two more regimes can theoretically exist. In the second, which lies between the upper concave curve and a line, the depletion is insufficient to regularize solutions, so no more than Leray's weak solutions exist. In the third, which lies above this line, solutions are regular, but correspond to extreme initial conditions. The paper ends with a discussion on the possibility of transition between these regimes.
Item Type: | Journal Article | ||||||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||
Journal or Publication Title: | Nonlinearity | ||||||||||
Publisher: | Institute of Physics Publishing Ltd. | ||||||||||
ISSN: | 0951-7715 | ||||||||||
Official Date: | October 2014 | ||||||||||
Dates: |
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Volume: | Volume 27 | ||||||||||
Number: | Number 10 | ||||||||||
Page Range: | pp. 2605-2625 | ||||||||||
DOI: | 10.1088/0951-7715/27/10/2605 | ||||||||||
Status: | Peer Reviewed | ||||||||||
Publication Status: | Published |
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