Large deviations for the stochastic shell model of turbulence

Abstract

Abstract. In this work, we first prove the existence and uniqueness of a strong solution to stochastic GOY model of turbulence with a small multiplicative noise. Then using the weak convergence approach, Laplace principle for solutions of the stochastic GOY model is established in certain Polish space. Thus a Wentzell-Freidlin type large deviation principle is established utilizing certain results by Varadhan and Bryc. Mathematics Subject Classification (2000). Primary 60F10; Secondary 60H15, 76D03, 76D06.

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

“…Thus, this result contains the corresponding existence and uniqueness theorems and a priori bounds for 2D Navier-Stokes equations (see, e.g. [28,34]), for the Boussinesq model of the Bénard convection (see [17], [14]), and also for the GOY shell model of turbulence (see [1] and [27]). Theorem 2.4 generalizes the existence result for MHD equations given in [2] to the case of multiplicative noise and also covers new situations such as the 2D magnetic Bénard problem, the 3D Leray α-model and the Sabra shell model of turbulence.…”

“…However, since we deal with an abstract hydrodynamical model with a forcing term which contains a stochastic control under a minimal set of hypotheses, the argument requires substantial modifications compared to that of [34] or [27]. It relies on a two-step Gronwall lemma (see Lemma 4.1 below and also [14]).…”

“…This is related to the Laplace principle. This approach has been already applied in several specific infinite dimensional situations (see, e.g, [34] for 2D Navier-Stokes equations, [14] for 2D Bénard convection, [4] for stochastic reactiondiffusion system, [25,31] for stochastic p-Laplacian equation and some its generalizations, [27] for the GOY shell model of turbulence). We also refer to [6] for large deviation results for evolution equations in the case on non-Lipschitz coefficients.…”

“…In particular, in addition to the 2D Navier-Stokes equations and the Boussinesq model mentioned above, Theorem 3.2 also proves LDP for 2D MHD equations, 2D magnetic Bénard convection, 3D Leray α-model, the Sabra shell model and dyadic model of turbulence. Note that unlike [34] and [27], in order to give a complete argument for the weak convergence (Proposition 3.4) and the compactness result (Proposition 3.5), we need to prove a time approximation result (Lemma 3.3). This requires to make stronger assumptions on the diffusion coefficient σ, which should have some Hölder time regularity, and in the explicit hydrodynamical models, no longer can include the gradient of the solution (see also [14]).…”

Stochastic magneto-hydrodynamic system perturbed by general noise

“…The Minty-Browder argument was used by us [15] to establish the existence and uniqueness of the two-dimensional Navier-Stokes system with a small multiplicative noise. In our earlier works such as [15] and [7], the objective was to prove the Freidlin-Wentzell type large deviations result for solutions of stochastic Navier-Stokes equations and certain shell models of turbulence.…”