On 2D Eulerian limits à la Kuksin

Abstract

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 −νΔv 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(−Δ)pv for p>0 and p≠1. We prove that the Eulerian limit process depends on the noise and on the parameter p; hence the Eulerian limits obtained for p≠1 are different from those obtained by Kuksin when p=1.