Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

Abstract

Abstract. We deal with a class of abstract nonlinear stochastic models, which covers many 2D hydrodynamical models including 2D Navier-Stokes equations, 2D MHD models and 2D magnetic Bénard problem and also some shell models of turbulence. We first prove the existence and uniqueness theorem for the class considered. Our main result is a Wentzell-Freidlin type large deviation principle for small multiplicative noise which we prove by weak convergence method.

Approximation of the stochastic 2D hydrodynamical type systems driven by non-Gaussian Lévy noise

“…It also covers the 3D Leray-α model for the Navier-Stokes equations and some shell models of turbulence. For details, we refer to ( [8], Sec. 2.1).…”

“…As it was explained in [8], Condition (A) covers many 2D hydrodynamical models including the 2D Navier-Stokes equations, the 2D magnetohydrodynamic equations, the 2D Boussinesq model of the Bénard convection, the 2D magnetic Bénard problem, the 3D Leray-α model for the Navier-Stokes equations and several shell models of turbulence.…”

Strong solutions for the stochastic 3D LANS-α model driven by non-Gaussian Lévy noise

“…Before starting the proof of our main results, we state the following version of the Gronwall Lemma whose proof can be found in [15]. Lemma 1.…”

Invariant Measures of Gaussian Type for 2D Turbulence

“…Here, we present the details for the SABRA shell model (see [24]), but the same results hold for the GOY shell model (see [21,25]). In recent years, there has been an increasing interest in these fluid dynamical models, both for the deterministic and the stochastic case (see also [3,5,9,10]). From the analytic point of view as well as for numerical computations, they are easier to analyze than the Navier-Stokes or Euler equations.…”