Statistical properties of nonlinear shell models of turbulence from linear advection models: rigorous results

Abstract

In a recent paper it was proposed that for some nonlinear shell models of turbulence one can construct a linear advection model for an auxiliary field such that the scaling exponents of all the structure functions of the linear and nonlinear fields coincide. The argument depended on an assumption of continuity of the solutions as a function of a parameter. The aim of this paper is to provide a rigorous proof for the validity of the assumption. In addition we clarify here when the swap of a nonlinear model by a linear one will not work,

Stochastic Attractors for Shell Phenomenological Models of Turbulence

“…σ n is a sequence of complex numbers, that are usually chosen equal to zero for all n greater than some n 0 (which describes the range of wavenumbers and consequently the length scales of external forces), (β n ) n≥1 is a sequence of independent complex valued Brownian motions. A rigorous theoretical analysis of the stochastic equation (1) and some of its statistical properties have been investigated in [4], while other rigorous results in the case of deterministic force have been developed in [5], [14], [15], [16]. The exact form of b n (·, ·) varies from one model to another.…”

“…Equation (3) should be considered as an auxiliary equation which, to some extent, may have similar statistical properties to those of equation (1), but is amenable to linear analysis (for instance the use of propagators). There is some numerical and heuristic evidence that some statistical properties of the solutions to equation (3), like the scaling exponents of the structure functions, are the same as those of the solutions to equation (1), see [1] and [5]. It is then of interest to understand the properties of the joint system du n + νk 2 n u n + b n (u, u) dt = σ n dβ n (4) dw n + νk 2 n w n + b n (u, w) dt = σ n dβ n , for n = 1, 2, ...…”

“…It is then of interest to understand the properties of the joint system du n + νk 2 n u n + b n (u, u) dt = σ n dβ n (4) dw n + νk 2 n w n + b n (u, w) dt = σ n dβ n , for n = 1, 2, ... In addition, the following idea has been introduced first in [1] and proved rigorously later in [5]: one can symmetrize system (4) by means of two additional terms as follows du n + νk 2 n u n + b n (u, u) + λb n (w, u) dt = σ n dβ n (5) dw n + νk 2 n w n + b n (u, w) + λb n (w, w) dt = σ n dβ n ,…”

“…where λ ∈ R is a parameter, and to analyze the dependence on λ of the properties of (5). For λ = 0 we recover (4).…”

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

“…It is then of interest to understand rigorously the properties of the joint system being m(du) = µ(du, dw) the unique invariant measure for (1). The analysis of system (6) has been performed by adding two terms (see [3] and the references therein):…”