Large deviations for the two-dimensional Navier–Stokes equations with multiplicative noise

Abstract

A Wentzell-Freidlin type large deviation principle is established for the two-dimensional Navier-Stokes equations perturbed by a multiplicative noise in both bounded and unbounded domains. The large deviation principle is equivalent to the Laplace principle in our function space setting. Hence, the weak convergence approach is employed to obtain the Laplace principle for solutions of stochastic Navier-Stokes equations. The existence and uniqueness of a strong solution to (a) stochastic Navier-Stokes equations with a small multiplicative noise, and (b) Navier-Stokes equations with an additional Lipschitz continuous drift term are proved for unbounded domains which may be of independent interest.

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.…”

“…[34] and [14]). This cannot be deduced from similar ones on u by means of a Girsanov transformation since the Girsanov density is not uniformly bounded when the intensity of the noise tends to zero (see [14]).…”

“…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]).…”