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

Hakima Bessaih; Benedetta  Ferrario

doi:10.3934/dcdsb.2016080

Article Contents

2016, Volume 21, Issue 9: 2927-2947. Doi: 10.3934/dcdsb.2016080

This issue Previous Article Preface Next Article Equi-attraction and continuity of attractors for skew-product semiflows

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

Hakima Bessaih^1, and
Benedetta Ferrario^2,

1.
University of Wyoming, Department of Mathematics, Dept. 3036, 1000 East University Avenue, Laramie, WY 82071
2.
Università di Pavia, Dipartimento di Matematica, via Ferrata 5, 27100 Pavia

Received: December 31, 2015

Revised: February 29, 2016

Published: October 2016

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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 $\lambda=0$.

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Mathematics Subject Classification: Primary: 60H15, 60G10; Secondary: 76D06.

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