Weak and strong solutions of the 3D Navier–Stokes equations and their relation to a chessboard of convergent inverse length scales
File(s)
Accepted version
Author(s)
Gibbon, JD
Type
Journal Article
Abstract
Using the scale invariance of the Navier–Stokes equations to define appropriate space-and-time-averaged inverse length scales associated with weak solutions of the 3D Navier–Stokes equations, on a periodic domain =[0,L]3 an infinite ‘chessboard’ of estimates for these inverse length scales is displayed in terms of labels (n,m) corresponding to n derivatives of the velocity field in L2m(). The (1,1) position corresponds to the inverse Kolmogorov length Re3/4. These estimates ultimately converge to a finite limit, Re3, as n,m→∞, although this limit is too large to lie within the physical validity of the equations for realistically large Reynolds numbers. Moreover, all the known time-averaged estimates for weak solutions can be rolled into one single estimate, labelled by (n,m). In contrast, those required for strong solutions to exist can be written in another single estimate, also labelled by (n,m), the only difference being a factor of 2 in the exponent. This appears to be a generalization of the Prodi–Serrin conditions for n≥1.
Date Issued
2019-02-01
Date Acceptance
2018-06-25
Citation
Journal of Nonlinear Science, 2019, 29 (1), pp.215-218
ISSN
0938-8974
Publisher
Springer Verlag
Start Page
215
End Page
218
Journal / Book Title
Journal of Nonlinear Science
Volume
29
Issue
1
Copyright Statement
© 2018 Springer-Verlag. The final publication is available at Springer via https://dx.doi.org/10.1007/s00332-018-9484-8.
Subjects
Science & Technology
Physical Sciences
Technology
Mathematics, Applied
Mechanics
Physics, Mathematical
Mathematics
Physics
Navier-stokes
Length scales
Weak and strong solutions
MOMENTS
TURBULENCE
Fluids & Plasmas
0102 Applied Mathematics
Publication Status
Published
Date Publish Online
2018-07-02