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 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(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.2000 Mathematics Subject Classification. 35Q30, 60H30, 76F05.

Sufficient Conditions for Dual Cascade Flux Laws in the Stochastic 2d Navier–Stokes Equations

“…Proof of (3.1c). Analogous to the proof of the 4/5 law in [6] (see also [33]), we use (3.1b) to prove (3.1c). We begin with the balance given by Lemma 2.9, which we write here for the reader's convenience as…”

“…In our previous work [6], we derived the weakest known sufficient condition to deduce the Kolmogorov 4/5 law for statistically stationary martingale solutions of the 3d Navier-Stokes equations. The condition is simply lim ν→0 νE u 2 L 2 = 0, which we refer to as weak anomalous dissipation.…”

“…with both inertial ranges) in the large box limit, we use the following assumption. Equivalently, the above conditions can be stated in terms of the energy/enstrophy balance, which makes Definition 1.2 look a little more like classical anomalous dissipation assumptions, in contrast to that used in [6].…”

“…Remark 1.11. As in [6], if one has the suitable statistical symmetries, the averages in x and/or n can be removed.…”

“…Proof of (3.1a). This case is the easiest and also the most similar to calculations in [6], so we will only sketch the main ideas. Recalling Lemma 2.6, we write the equation for the spherically averaged fluxD(ℓ) asD…”