A sufficient condition for the Kolmogorov 4/5 law for stationary martingale solutions to the 3D Navier-Stokes equations
File(s)
Accepted version
Author(s)
Bedrossian, Jacob
Zelati, Michele Coti
Punshon-Smith, Samuel
Weber, Franziska
Type
Journal Article
Abstract
We prove that statistically stationary martingale solutions of the 3D Navier-Stokes equations on 3 subjected to white-in-time (colored-in-space) forcing satisfy the Kolmogorov 4/5 law (in an averaged sense and over a suitable inertial range) using only the assumption that the kinetic energy is o(ν−1) as ν→0 (where ν is the inverse Reynolds number). This plays the role of a weak anomalous dissipation. No energy balance or additional regularity is assumed (aside from that satisfied by all martingale solutions from the energy inequality). If the force is statistically homogeneous, then any homogeneous martingale solution satisfies the spherically averaged 4/5 law pointwise in space. An additional hypothesis of approximate isotropy in the inertial range gives the traditional version of the Kolmogorov law. We demonstrate a necessary condition by proving that energy balance and an additional quantitative regularity estimate as ν→0 imply that the 4/5 law (or any similar scaling law) cannot hold.
Date Issued
2019-05
Date Acceptance
2018-12-27
Citation
Communications in Mathematical Physics, 2019, 367 (3), pp.1045-1075
ISSN
0010-3616
Publisher
Springer (part of Springer Nature)
Start Page
1045
End Page
1075
Journal / Book Title
Communications in Mathematical Physics
Volume
367
Issue
3
Copyright Statement
© 2019 Springer-Verlag GmbH Germany, part of Springer Nature. The final publication is available at Springer via https://doi.org/10.1007/s00220-019-03396-6
Identifier
http://arxiv.org/abs/1803.09695v1
Subjects
math.AP
math.AP
physics.flu-dyn
Publication Status
Published
Date Publish Online
2019-03-11