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

Abstract

We show that the system of point vortices, perturbed by a certain transport type noise, converges weakly to the vorticity form of 2D Navier-Stokes equations driven by the spacetime white noise.

A Central Limit Theorem for Gibbsian Invariant Measures of 2D Euler Equations

“…and approximation by point vortices systems, [13,14,15,16]. The latter ones are defined, let us say first on T 2 , as systems of N point particles with positions x i ∈ D and intensities ξ i ∈ R, satisfying the system of ordinary differential equations ẋi,t = − j =i ξ j ∇ ⊥ G(x i,t , x j,t ),…”

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

“…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.…”