Large, moderate deviations principle and $α$-limit for the 2D Stochastic LANS-$α$

Abstract

In this paper we consider the Lagrangian Averaged Navier-Stokes Equations, also known as, LANS-α Navier-Stokes model on the two dimensional torus. We assume that the noise is cylindrical Wiener process and its coefficient is multiplied by √ α. We then study through the lenses of the large and moderate deviations principle the behaviour of the trajectories of the solutions of the stochastic system as α goes to 0. We first show that as α goes to 0, the solutions of the stochastic LANS-α converge in probability to the solutions of the deterministic Navier-Stokes equations. Then, we present a unifying approach to the proof of the two deviations principles and express the rate function in term of the solution of the Navier-Stokes equations. Our proof is based on the weak convergence approach to large deviations principle.