Istituto di Matematica Applicata e Tecnologie Informatiche     
Ferrario B. Stochastic Navier-Stokes equations: Analysis of the noise to have a unique invariant measure. Preprint ercim.cnr.ian//1997-1053, 1997.
The aim of the paper is to provide a complete analysis of sufficient conditions for the existence of a unique invariant measure for the 2D Navier--Stokes equations perturbed by a white noise term. An invariant measure is, roughly speaking, a ``statistical stationary solution'' and is a good candidate to represent the asymptotic behavior of the system. If this invariant measure is unique, there are chances that the law of the process solution will converge to it. Therefore, when this holds true, this unique invariant measure describes the (statistical) equilibrium to which the system tends. In this paper we prove that a unique invariant measure exists and the convergence takes place when the 2D Navier-Stokes equations are perturbed by a time--white noise, not degenerate in space, but with no limitations on the way it affects the modes of the phase space. From the technical point of view, this means to check irreducible and strongly Feller properties; such properties characterize the asymptotic behavior of general semilinear stochastic evolution equations and moreover are sufficient conditions for the uniqueness of the invariant measure. In general, without constraints on the Reynolds number, the deterministic Navier-Stokes equations have many stationary solutions. No information about the long time behavior is directly related to them. But they can be understood as invariant measures for the Navier-Stokes equations without the noise. Hence, our result means that, when a sufficiently distributed random perturbation is added, just one invariant measure exists. The effect of the noise is to mix up the dynamics of the system, allowing a unique asymptotic behavior.
Subject 65-XX

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