Large, moderate deviations principle and α\alphaα-limit for the 2D Stochastic LANS-$\alpha$ (original) (raw)

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

2021

In this paper we consider the Lagrangian Averaged Navier-Stokes Equations, also known as, LANS-α Navier-Stokes model on the two dimensional torus. We assume that the noise is a cylindrical Wiener process and its coefficient is multiplied by √(α). We then study through the lenses of the large and moderate deviations principle the behaviour of the trajectories of the solutions of the stochastic system as α 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-α model converge in probability to the solutions of the deterministic Navier-Stokes equations.

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

SIAM Journal on Mathematical Analysis, 2012

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.

Inviscid Large deviation principle and the 2D Navier Stokes equations with a free boundary condition

2012

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.

Probabilistic estimates for the Two Dimensional Stochastic Navier-Stokes Equations

Journal of Statistical Physics, 2000

We consider the Navier-Stokes equation on a two dimensional torus with a random force, white noise in time and analytic in space, for arbitrary Reynolds number RRR. We prove probabilistic estimates for the long time behaviour of the solutions that imply bounds for the dissipation scale and energy spectrum as RtoinftyR\to\inftyRtoinfty.

Ergodicity for stochastic equation of Navier--Stokes type

arXiv: Probability, 2020

In the first part of the note we analyze the long time behaviour of a two dimensional stochastic Navier-Stokes equation (N.S.E.) system on a torus with a degenerate, one dimensional noise. In particular, for some initial data and noises we identify the invariant measure for the system and give a sufficient condition under which it is unique and stochastically stable. In the second part of the note, we consider a simple example of a finite dimensional system of stochastic differential equations driven by a one dimensional Wiener process with a drift, that displays some similarity with the stochastic N.S.E., and investigate its ergodic properties depending on the strength of the drift. If the latter is sufficiently small and lies below a critical threshold, then the system admits a unique invariant measure which is Gaussian. If, on the other hand, the strength of the noise drift is larger than the threshold, then in addition to a Gaussian invariant measure, there exist another one. In...

On Inviscid Limits for the Stochastic Navier–Stokes Equations and Related Models

Archive for Rational Mechanics and Analysis, 2015

We study inviscid limits of invariant measures for the 2D Stochastic Navier-Stokes equations. As shown in [Kuk04] the noise scaling ν is the only one which leads to non-trivial limiting measures, which are invariant for the 2D Euler equations. We show that any limiting measure µ 0 is in fact supported on bounded vorticities. Relationships of µ 0 to the long term dynamics of Euler in the L ∞ with the weak * topology are discussed. In view of the Batchelor-Krainchnan 2D turbulence theory, we also consider inviscid limits for the weakly damped stochastic Navier-Stokes equation. In this setting we show that only an order zero noise (i.e. the noise scaling ν 0 ) leads to a nontrivial limiting measure in the inviscid limit.

Stochastic 2D hydrodynamical type systems: Well posedeness and large deviations

HAL (Le Centre pour la Communication Scientifique Directe), 2010

We deal with a class of abstract nonlinear stochastic models, which covers many 2D hydrodynamical models including 2D Navier-Stokes equations, 2D MHD models and the 2D magnetic Bénard problem and also some shell models of turbulence. We state the existence and uniqueness theorem for the class considered. Our main result is a Wentzell-Freidlin type large deviation principle for small multiplicative noise which we prove by a weak convergence method. Keywords Hydrodynamical models • MHD • Bénard convection • Shell models of turbulence • Stochastic PDEs • Large deviations The research of the second named author is partially supported by the research project BMF2003-01345.

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

Applied Mathematics and Optimization, 2010

We deal with a class of abstract nonlinear stochastic models, which covers many 2D hydrodynamical models including 2D Navier-Stokes equations, 2D MHD models and 2D magnetic Bénard problem and also some shell models of turbulence. We first prove the existence and uniqueness theorem for the class considered. Our main result is a Wentzell-Freidlin type large deviation principle for small multiplicative noise which we prove by weak convergence method.