A sufficient condition for the Kolmogorov 4/5 law for stationary martingale solutions to the 3D Navier-Stokes equations

Abstract

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 $\nu \to 0$ (where $\nu$ 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 $\nu \to 0$ imply that the 4/5 law (or any similar scaling law) cannot hold.

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Cited by in corpus (6)

• Self-Regularization in turbulence from the Kolmogorov 4/5-Law and Alignment
• Three-dimensional shear driven turbulence with noise at the boundary
• Sufficient conditions for dual cascade flux laws in the stochastic 2d Navier-Stokes equations
• Dependence of the asymptotic energy dissipation on third-order velocity scaling
• Sufficient conditions for local scaling laws for stationary martingale solutions to the 3D Navier-Stokes equations
• Kolmogorov $4/5$ law for the forced 3D Navier-Stokes equations