Statistical properties of stochastic 2D Navier-Stokes equations from linear models
Article
Bessaih, H, Ferrario, B. (2016). Statistical properties of stochastic 2D Navier-Stokes equations from linear models
. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, 21(9), 2927-2947. 10.3934/dcdsb.2016080
Bessaih, H, Ferrario, B. (2016). Statistical properties of stochastic 2D Navier-Stokes equations from linear models
. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, 21(9), 2927-2947. 10.3934/dcdsb.2016080
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 λ which gives a system (uλ, wλ); this system is studied for any λ proving its well posedness and the uniqueness of its invariant measure μλ. The key point is that for any λ ≠ 0 the fields uλ and wλ 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 λ → 0, proving that μλ weakly converges to μ0, where μ0 is the only invariant measure for the joint system for (u, w) when λ = 0.
