The Eulerian Limit for 2D Statistical Hydrodynamics

We consider the 2D Navier–Stokes system, perturbed by a white in time random force, proportional to the square root of the viscosity. We prove that under the limit “time to infinity, viscosity to zero” each of its (random) solution converges in distribution to a non-trivial stationary process, formed by solutions of the (free) Euler equation, while the Reynolds number grows to infinity. We study the convergence and the limiting solutions.