Communications in Mathematical Physics

, Volume 367, Issue 3, pp 1045–1075 | Cite as

A Sufficient Condition for the Kolmogorov 4/5 Law for Stationary Martingale Solutions to the 3D Navier–Stokes Equations

  • Jacob Bedrossian
  • Michele Coti Zelati
  • Samuel Punshon-SmithEmail author
  • Franziska Weber


We prove that statistically stationary martingale solutions of the 3D Navier–Stokes equations on \({\mathbb{T}^3}\) 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 \({o(\nu^{-1})}\) as \({\nu \rightarrow 0}\) (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 \({\nu \rightarrow 0}\) imply that the 4/5 law (or any similar scaling law) cannot hold.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA
  3. 3.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  4. 4.Department of MathematicsImperial College LondonLondonUK

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