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

Abstract

We provide sufficient conditions for mathematically rigorous proofs of the third order universal laws capturing the energy flux to large scales and enstrophy flux to small scales for statistically stationary, forced-dissipated 2d Navier-Stokes equations in the large-box limit. These laws should be regarded as 2d turbulence analogues of the 4/5 law in 3d turbulence, predicting a constant flux of energy and enstrophy (respectively) through the two inertial ranges in the dual cascade of 2d turbulence. Conditions implying only one of the two cascades are also obtained, as well as compactness criteria which show that the provided sufficient conditions are not far from being necessary. The specific goal of the work is to provide the weakest characterizations of the "0-th laws" of 2d turbulence in order to make mathematically rigorous predictions consistent with experimental evidence. Contents2010 Mathematics Subject Classification. 35Q30, 60H30, 76F05.

Sufficient conditions for local scaling laws for stationary martingale solutions to the 3D Navier-Stokes equations

“…However, see [2] for a proof of a weak anomalous dissipation condition for the problem of passive scalar advection. At any rate, provided that one can obtain relations similar to the ones we derive in the next section, we believe that similar steps can be taken to derive local cascade laws for other systems that are expected to exhibit this sort of behavior, for instance one could attempt to adapt the derivations of the scaling laws in [4] or [2] to bounded domains in the same way that our paper adapts the results in [3].…”

Lagrangian chaos and scalar advection in stochastic fluid mechanics

“…Remark 1. 18. Note that by the balance (1.6), the weak anomalous dissipation property (1.7), and Sobolev interpolation, there holds lim Ä!0 ÄEkg Ä k 2 H D 0 for all 2 .0; 1/ and lim Ä!0 ÄEkg Ä k 2 H D C1 for all > 1.…”

“…19. There exist a number of works providing sufficient conditions to deduce turbulence scaling laws and related results, see for example [17,18,51] in the stochastic case and [31,42,47,93] in deterministic cases. See the references therein for earlier work in the physics literature.…”

“…Thankfully, condition (C) is far from necessary to rule out the criterion in Theorem 3. 18. In this section we prove a sufficient condition, weaker than (C), which is better suited for infinite-dimensional RDS.…”

“…Theorem 3. 18. If C D , then for each z 2 Z there a is Borel measure z on P d 1 such that (i) the assignment z 7 !…”