Large Deviations and the Zero Viscosity Limit for 2D Stochastic Navier–Stokes Equations with Free Boundary

Abstract

Using a weak convergence approach, we prove a LPD for the solution of 2D stochastic Navier Stokes equations when the viscosity converges to 0 and the noise intensity is multiplied by the square root of the viscosity. Unlike previous results on LDP for hydrodynamical models, the weak convergence is proven by tightness properties of the distribution of the solution in appropriate functional spaces.

On stochastic modified 3D Navier–Stokes equations with anisotropic viscosity

“…Examples of such coefficients σ are provided. Since we are working on the whole space R 3 , and not on a bounded domain, the martingale approach used in [4], which depends on tightness properties, does not seem appropriate. We use instead the control method introduced in [20] for the 2D Navier-Stokes equation; see also [26], [15], [13] and [24], where this method has been used for the stochastic 2D Navier-Stokes equations, stochastic 2D general hydrodynamical Bénard models and the stochastic 3D tamed Navier-Stokes equations.…”

“…Note that as usual, starting with an initial condition u 0 ∈H 0,1 and projecting equation (1.2) on the space of divergence-free fields, we get rid of the pressure and rewrite the evolution equation as follows: 4) and P div denotes the projection on divergence free functions. For u ∈ H 1 such that ∇ · u = 0, set B(u) := B(u, u).…”

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

“…We refer, for instance, to [29] and [35] for more detailed explanation and historical account of the MDP. We refer, for instance, to [4], [2], [3], [12], [64], [65] [5], [6], [12], [19], [37], [39], [62], [61], [60] and references therein for a small sample of results from the extensive literature devoted to MDP and LDP for stochastic differential equations with small noise.…”

Large deviations for stochastic tidal dynamics equation

“…Rockner et al [35] studied the large deviations for the stochastic tamed 3D Navier-Stokes equations, whilst Cheushov and Millet [13] have considered the 2D hydrodynamical type systems which includes 2D Navier-Stokes equation as a special case. An LDP for 2D Stochastic Navier-Stokes equation with free boundary was discussed by Bessaih and Millet [3]. The large deviations for a stochastic Burgers' equation was established by [37] using the weak convergence approach.…”