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# Communications in Mathematical Sciences

## Volume 20 (2022)

### Number 8

### Large, moderate deviations principle and $\alpha$-limit for the 2D stochastic LANS-$\alpha$

Pages: 2231 – 2264

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n8.a5

#### Authors

#### Abstract

In this paper we consider the Lagrangian Averaged Navier–Stokes Equations, also known as, LANS‑$\alpha$ Navier–Stokes model on the two dimensional torus. We assume that the noise is a cylindrical Wiener process and its coefficient is multiplied by $\sqrt{\alpha}$. We then study through the lenses of the large and moderate deviations principles the behaviour of the trajectories of the solutions of the stochastic system as $\alpha$ goes to $0$. Instead of giving two separate proofs of the two deviations principles we present a unifying approach to the proof of the LDP and MDP and express the rate function in term of the unique solution of the Navier–Stokes equations. Our proof is based on the weak convergence approach to large deviations principle. As a by-product of our analysis we also prove that the solutions of the stochastic LANS‑$\alpha$ model converge in probability to the solutions of the deterministic Navier–Stokes equations.

#### Keywords

LANS-$\alpha$ model, Camassa–Holm equations, large deviation principle, stochastic Navier–Stokes equations

#### 2010 Mathematics Subject Classification

35R60, 60F10, 76D05

Received 2 August 2021

Received revised 6 February 2022

Accepted 12 March 2022

Published 29 November 2022