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 by Kuksin (J Stat Phys 115(1–2):469–492, 2004), the noise scaling $${\sqrt{\nu}}$$ν is the only one which leads to non-trivial limiting measures, which are invariant for the 2D Euler equations. Using a Moser-type iteration for stochastic drift-diffusion equations, we show that any limiting measure $${\mu_{0}}$$μ0 is in fact supported on bounded vorticities. Relationships of $${\mu_{0}}$$μ0 to the long term dynamics of 2D Euler in $${L^{\infty}}$$^L∞ with the weak^* topology are discussed. We also obtain a drift-independent modulus of continuity for a stationary deterministic model problem, which leads us to conjecture that in fact $${\mu_0}$$μ0 is supported on $${C^0}$$^C0. Moreover, in view of the Batchelor–Krainchnan 2D turbulence theory, we consider inviscid limits for a weakly damped stochastic Navier–Stokes equation. In this setting we show that only an order zero noise scaling (with respect to ν) leads to a nontrivial limiting measure in the inviscid limit.