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 $L^{\mathrm{\infty }}$ with the weak${}^{\ast }$ 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.