Continuity equation in LlogL for the 2D Euler equations under the enstrophy measure

Abstract

The 2D Euler equations with random initial condition has been investigates by S. Albeverio and A.-B. Cruzeiro in [1] and other authors. Here we prove existence of solutions for the associated continuity equation in Hilbert spaces, in a quite general class with LlogL densities with respect to the enstrophy measure.

Fokker-Planck equation for dissipative 2D Euler equations with cylindrical noise

“…The Euler nonlinear term in the Gaussian setting. The definition of the nonlinear term in (2.1) when the law of ω t is µ, or an absolutely continuous measure with respect to µ, is not immediate, and it has been thoroughly discussed in [33] and related works, [35,26,45]. We will rely upon the arguments of Subsection 2.5 of [33], which we now review.…”

“…Thus, the second term on the right hand side tends to 0 as i → ∞. Next, one can prove that b N i , Dψ converge strongly in L 1 (E, µ) to B, Dψ as i → ∞, see for instance[26, Section 3.3.1]. Combining the convergence with the uniform exponential integrability of these quantities, we deduce that the sequence b N i , Dψ actually converges to B, Dψ in the Orlicz norm.…”

“…To give a rigorous definition of the Liouville operator B of Euler equation (1.2), we make use of Proposition 2.1 (see also the discussion in [26]). First, we combine the latter two expressions with (2.4) to obtain…”

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.…”