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

TL;DR

This paper establishes sufficient conditions under which local scaling laws of turbulence, like the 4/5 and 4/3 laws, hold for stationary martingale solutions of the 3D Navier-Stokes equations in bounded domains.

Contribution

It introduces a precompactness condition in L^3 that ensures local structure functions relate to fluxes and dissipation in the vanishing viscosity limit.

Findings

Local structure functions are expressed as nonlinear fluxes at small scales.

Precompactness in L^3 leads to solutions of Euler equations with similar forcing.

Conditions are identified under which local turbulence laws hold in bounded domains.

Abstract

The main goal of this paper is to obtain sufficient conditions that allow us to rigorously derive local versions of the 4/5 and 4/3 laws of hydrodynamic turbulence, by which we mean versions of these laws that hold in bounded domains. This is done in the context of stationary martingale solutions of the Navier-Stokes equations driven by an Ornstein-Uhlenbeck process. Specifically, we show that under an assumption of \say{on average} precompactness in $L^3,$ the local structure functions are expressed up to first order in the length scale as nonlinear fluxes, in the vanishing viscosity limit and within an appropriate range of scales. If in addition one assumes local energy equality, this is equivalent to expressing the structure functions in terms of the local dissipation. Our precompactness assumption is also shown to produce stationary martingale solutions of the Euler equations with the same type of forcing in the vanishing viscosity limit.