On 2D Eulerian limits à la Kuksin

We prove the existence of stochastic processes solving the deterministic Euler equations for an inviscid fluid on the 2D torus. In [20] Kuksin obtained this result by approximating the Euler equations by the stochastic Navier-Stokes equations with viscous term −ν\Delta v and intensity of the noise vanishing as √ν; then in the limit as ν → 0 non trivial stationary processes solving the deterministic Euler equations were obtained. In this paper we modify the approximating viscous equations by considering a dissipative term ν(−\Delta)^pv for p > 0 and p\neq 1. We prove that the Eulerian limit process depends on the noise and on the parameter p; hence the Eulerian limits obtained for p\neq 1 are different from those obtained by Kuksin when p = 1.