On inviscid limits for the stochastic Navier-Stokes equations and related

Abstract

ABSTRACT. We study inviscid limits of invariant measures for the 2D Stochastic Navier-Stokes equations. As shown in [Kuk04] the noise scaling p 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 µ0 is in fact supported on bounded vorticities. Relationships of µ0 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 ν0) leads to a nontrivial limiting measure in the inviscid limit. 1