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We study inviscid limits of invariant measures for the 2D StochasticNavier-Stokes equations. As shown in \cite{Kuksin2004} the noise scaling$\sqrt{{\nu}}$ is the only one which leads to non-trivial limiting measures,which are invariant for the 2D Euler equations. We show that any limitingmeasure $\mu_{0}$ is in fact supported on bounded vorticities. Relationships of$\mu_{0}$ to the long term dynamics of Euler in the $L^{\infty}$ with theweak$^{*}$ topology are discussed. In view of the Batchelor-Krainchnan 2Dturbulence theory, we also consider inviscid limits for the weakly dampedstochastic Navier-Stokes equation. In this setting we show that only an orderzero noise (i.e. the noise scaling $\nu^0$) leads to a nontrivial limitingmeasure in the inviscid limit.

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