Viscosity limit and deviations principles for a grade-two fluid driven by multiplicative noise

Abstract

In this paper we study a grade-two fluid driven by multiplicative Gaussian noise. Under appropriate assumptions on the initial condition and the noise, we prove a large and moderate deviations principle in the space Cpr0, T s; H m q, m P t2, 3u, of the solution of our stochastic model as the viscosity ε converges to 0 and the coefficient of the noise is multiplied by ε 1 2. 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 inviscid grade-two fluid which is also known as Lagrangian Averaged Euler equations. Our proof is based on the weak convergence approach to large deviations principle.

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

“…In this present paper, instead of presenting two separate proofs of the LDP and MDP results we present a unifying approach for these deviation principles for the LANS-α model. A similar approach was introduced in [34] for the vanishing viscosity limit of the second grade fluid. To be precise, we fix δ ∈ {0, 1} and consider the following problem          dy α,δ + Ay α,δ + λ δ (α) Bα (y α,δ , z α,δ ) + δ Bα (u, z α,δ ) + Bα (y α,δ , J −1 α u)…”

Well-posedness of stochastic third grade fluid equation

“…However, for the stochastic equation, due to the lack of regularity with respect to time and to the stochastic parameter, we can not use the compactness arguments to pass to the limit in those nonlinear terms. Instead, we will follow the methods introduced in [4], which have been successfully applied to the stochastic second grade fluids in [8], [21] (see also [9], [10], [13], [19], [22], [24], [25]). More precisely, we consider an appropriate Galerkin basis, and deduce suitable uniform estimates in order to get weak convergence of a subsequence.…”