Statistical properties of stochastic 2D Navier-Stokes equations from linear models

TL;DR

This paper explores the statistical properties of the 2D Navier-Stokes equations driven by noise, proposing a linear model approach to understand the anomalous scaling exponents of turbulence and analyzing the convergence of invariant measures.

Contribution

It introduces a linear passive advection model for the 2D Navier-Stokes equations and proves the convergence of invariant measures as a parameter approaches zero.

Findings

The coupled Navier-Stokes/linear advection system is well-posed.

Invariant measures are unique for each parameter value.

The invariant measures converge as the parameter tends to zero.

Abstract

A new approach to the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence has been proposed and tested through numerical simulations. This is achieved by constructing, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem. In this paper, we investigate this conjecture for the 2D Navier-Stokes equations driven by an additive noise. In order to check this conjecture, we analyze the coupled system Navier-Stokes/linear advection system in the unknowns $(u,w)$. We introduce a parameter $\lambda$ which gives a system $(u^\lambda,w^\lambda)$; this system is studied for any $\lambda$ proving its well posedness and the uniqueness of its invariant measure $\mu^\lambda$. The key point is that for any $\lambda \neq 0$ the fields $u^\lambda$ and $w^\lambda$ have the same scaling exponents, by assuming universality of the scaling exponents to the force. In order to prove the same for the original fields $u$ and $w$, we investigate the limit as $\lambda \to 0$, proving that $\mu^\lambda$ weakly converges to $\mu^0$, where $\mu^0$ is the only invariant measure for the joint system for $(u,w)$ when $\lambda=0$.