Probabilistic estimates for the Two Dimensional Stochastic Navier-Stokes Equations (original) (raw)
Dynamics of stochastic 2D Navier–Stokes equations
Journal of Functional Analysis, 2010
In this paper, we study the dynamics of a two-dimensional stochastic Navier-Stokes equation on a smooth domain, driven by linear multiplicative white noise. We show that solutions of the 2D Navier-Stokes equation generate a perfect and locally compacting C 1,1 cocycle. Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-random Lyapunov spectrum for the cocycle. The Lyapunov spectrum characterizes the asymptotics of the cocycle near an equilibrium/stationary solution. We give sufficient conditions on the parameters of the Navier-Stokes equation and the geometry of the planar domain for hyperbolicity of the zero equilibrium, uniqueness of the stationary solution (viz. ergodicity), local almost sure asymptotic stability of the cocycle, and the existence of global invariant foliations of the energy space.
Exponential Mixing of the 2D Stochastic Navier-Stokes Dynamics
Communications in Mathematical Physics, 2002
We consider the Navier-Stokes equation on a two dimensional torus with a random force which is white noise in time, and excites only a finite number of modes. The number of excited modes depends on the viscosity nu\nunu, and grows like nu−3\nu^{-3}nu−3 when nu\nunu goes to zero. We prove that this Markov process has a unique invariant measure and is exponentially mixing in time.
Probability Theory and Related Fields, 2014
We are dealing with the Navier-Stokes equation in a bounded regular domain D of R 2 , perturbed by an additive Gaussian noise ∂w Q δ /∂t, which is white in time and colored in space. We assume that the correlation radius of the noise gets smaller and smaller as δ 0, so that the noise converges to the white noise in space and time. For every δ > 0 we introduce the large deviation action functional S δ 0,T and the corresponding quasi-potential U δ and, by using arguments from relaxation and Γ-convergence we show that U δ converges to U = U 0 , in spite of the fact that the Navier-Stokes equation has no meaning in the space of square integrable functions, when perturbed by space-time white noise. Moreover, in the case of periodic boundary conditions the limiting functional U is explicitly computed.
Stochastic Navier-Stokes Equations
Acta Applicandae Mathematicae, 1995
The purpose of this article is to survey some results related to the theory of stochastic Navier-Stokes equations (SNSE). The interest of SNSE arises from modelling turbulence. We begin to show how SNSE can be introduced intuitively from the random motion of particles. We then review briefly the deterministic theory and present the main core of existence theory for NSE. We also discuss uniqueness issues. We end up by showing how the splitting-up method provides a useful constructive approach to existence, and by presenting some extensions, like weakening assumptions or considering the special case of small initial data.
Stochastic 2D Navier-Stokes Equation and Applications to 2D Turbulence
2016
Author(s): Karimi, Shahab | Advisor(s): Birnir, Bjorn | Abstract: We will consider the 2-dimensional Navier-Stokes equation for an incompressible fluid with periodic boundary condition, and with a random perturbation that is in the form of white noise in time and a deterministic perturbation due to the large deviation principle. Our ultimate goal is to find appropriate conditions on the initial data and the forcing terms so that global existence and uniqueness of a mild solution is guaranteed. We will use the Picard's iteration method to prove existence of local mild solution and then prove the existence of a maximal solution which then leads to global existence. The result is applied to the backward Kolmogorov-Obukhov energy cascade and the forward Kraichnan enstrophy cascade in 2D turbulence.
The Stochastic 2D Navier—Stokes Equation
Kolmogorov Equations for Stochastic PDEs, 2004
In this paper we study the stochastic Navier-Stokes equation with artificial compressibility. The main results of this work are the existence and uniqueness theorem for strong solutions and the limit to incompressible flow. These results are obtained by utilizing a local monotonicity property of the sum of the Stokes operator and the nonlinearity.
A note on stochastic Navier-Stokes equations with not regular multiplicative noise
arXiv (Cornell University), 2015
We consider the Navier-Stokes equations in R d (d = 2, 3) with a stochastic forcing term which is white noise in time and coloured in space; the spatial covariance of the noise is not too regular, so Itô calculus cannot be applied in the space of finite energy vector fields. We prove existence of weak solutions for d = 2, 3 and pathwise uniqueness for d = 2.
Stochastic Navier-Stokes equations: Analysis of the noise to have a unique invariant measure
Annali di Matematica Pura ed Applicata, 1999
, it has been proven that there exists a unique invariant measure for the 2D Navier-Stokes equations perturbed by a white noise term; this is the probability measure representing the asymptotic behavior. There, the assumptions on the noise were quite restrictive. In this paper we remove the heaviest limitation, that is the lower bound on the range of the noise covariance, providing a complete analysis of sufficient conditions for the existence of a unique invariant measure.
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.
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...