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

Abstract

After a short review of recent progresses in 2D Euler equations with random initial conditions and noise, some of the recent results are improved by exploiting a priori estimates on the associated infinite dimensional Fokker-Planck equation. The regularity class of solutions investigated here does not allow energyor enstrophy-type estimates, but only bounds in probability with respect to suitable distributions of the initial conditions. This is a remarkable application of Fokker-Planck equations in infinite dimensions. Among the example of random initial conditions we consider Gibbsian measures based on renormalized kinetic energy.

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.…”

“…As in the case of 2-dimensional Euler's equations in the Energy-Enstrophy stationary regime, or more generally when fixed time marginals are absolutely continuous with respect to space white noise, see [13], uniqueness remains an important open problem. We will not discuss uniqueness of solutions of (BQG) in the above stationary regime; thus, in particular, we are not able to state that the solutions we produce form a flow, i.e.…”