Kolmogorov equations associated to the stochastic 2D Euler equations

Abstract

The Kolmogorov equation associated to a stochastic 2D Euler equations with transport type noise and random initial conditions is studied by a direct approach, based on Fourier analysis, Galerkin approximation and Wiener chaos methods. The method allows us to generalize previous results and to understand the role of the regularity of the noise, in relation to a limiting value of roughness.

Convergence of transport noise to Ornstein–Uhlenbeck for 2D Euler equations under the enstrophy measure

“…In the sense described in [2,11], the enstrophy measure µ is invariant also for this equation (for every β > 0, in this case). The same fact holds for a stochastic version of 2D Euler equations, but with transport type noise, as described in [12,13]:…”

“…where Z 2 0 = Z 2 \ {0} and e k (x) is the orthonormal basis of sine and cosine functions, see (2.1) below. In [12,13] the problem has been studied for γ > 2.…”

Equilibrium Statistical Mechanics of Barotropic Quasi-Geostrophic Equations

“…The following result finds analogues in [2, Lemma 1.3.2], see also [1], and in [12, Theorem 8] or the related [9,14,16,13], all dealing with stationary solutions of 2-dimensional Euler equations.…”

Point vortex approximation for 2D Navier--Stokes equations driven by space-time white noise

“…The proof of the above assertion follows the idea of [10,Appendix 6], with some combinatorial flavor here. Since l, m are fixed, we write R N instead of R l,m (ω N 0 ) for simplicity.…”