Stationary Solutions of Damped Stochastic 2-dimensional Euler's Equation

Abstract

Existence of stationary point vortices solution to the damped and stochastically driven Euler's equation on the two dimensional torus is proved, by taking limits of solutions with finitely many vortices. A central limit scaling is used to show in a similar manner the existence of stationary solutions with white noise marginals.

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

“…As a consequence, K(x, y) = K(x−y) is a translation invariant vector field, exploding at 0 with the speed |x| −1 . The forcing term dW is the space-time white noise on T 2 , so that our model coincides with the one studied in [45], where existence of weak (both in probabilistic and analytic sense) stationary solutions were proved by approximation with a system of Euler point vortices with creation and quenching.…”

“…The space-time white noise, a centred delta-correlated Gaussian field, is equivalently understood as the cylindrical Wiener process on L 2 (T 2 ), see [27]. The stationary fixed time marginal considered in [45] is the unique invariant measure of the linear part of the equation…”

“…In the stream of aforementioned works on stochastic 2D Euler equations with damping, but in the specific regime of distributional solutions -corresponding to the case of cylindrical white noise -investigated by [45], we extend here that result from stationary to non-stationary solutions, with random initial conditions related to a Gaussian invariant measure. As already remarked, this extension is based on preliminary estimates on the associated Fokker-Planck equation, where some technical aspects are inspired by recent works on a different kind of noise, see [36].…”

“…As for problem (i), a huge effort has been devoted to the attempt at extending and improving the deterministic theory by means of probability and stochastic models. The present work stems from the study of models proposed to describe features of turbulence such as the inverse energy cascade in dimension 2, see [16,45], and it may fit into the framework of question (i), since it extends the class of function spaces in which Euler equation is solved.…”

“…This theory was recently revised by means of an alternative approach based on point vortex approximation [33]. These works, devoted to the deterministic equation (1.2) with random initial conditions, have been generalized to stochastic cases, on one hand to the case of multipicative transport noise, see in particular [36,34,38]; on the other hand to the case of additive spacetime white noise and friction [45] (multiplicative noise is formally conservative, while in the case of additive noise a friction is needed to allow stationary solutions). The 2D Euler equations with additive noise, possibly including friction, their corresponding stationary solutions and invariant measures had already been considered before.…”