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

We study inviscid limits of invariant measures for the 2D Stochastic Navier-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 limiting measure $\mu_{0}$ is in fact supported on bounded vorticities. Relationships of $\mu_{0}$ to the long term dynamics of Euler in the $L^{\infty}$ with 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 $\nu^0$) leads to a nontrivial limiting measure in the inviscid limit.