Limit Theorems and Fluctuations for Point Vortices of Generalized Euler Equations

Abstract

We prove a mean field limit, a law of large numbers and a central limit theorem for a system of point vortices on the 2D torus at equilibrium with positive temperature. The point vortices are formal solutions of a class of equations generalising the Euler equations, and are also known in the literature as generalised inviscid SQG. The mean-field limit is a steady solution of the equations, the CLT limit is a stationary distribution of the equations.

Gaussian fluctuations around limit measures of generalized SQG point vortices

“…For the proof, we refer to [17]. Note that this theorem as well as the other results are asymptotic both in the number of vortices and the regularization parameter ǫ, and thus they capture the behavior of the original system.…”

“…The following result, taken from [17], Theorem 3.2., states a Law of Large Numbers for the joint empirical distribution. In terms of the vorticity θ, this result says that the limit is a stationary solution of the original equation.…”

“…The operator G provides the solution to the problem (−∆) m 2 Φ = φ with periodic boundary conditions and zero spatial average, and extends naturally to functions depending on both variables γ, x by acting on the spatial variable only. We state now the Central limit Theorem, and refer to [17], Theorem 3.4. for the proof.…”

“…To avoid this, periodic boundary conditions, i.e. domains such as the disc in [3] or the two-dimensional torus in [20] and [17] were used.…”

Linearization and a superposition principle for deterministic and stochastic nonlinear Fokker-Planck-Kolmogorov equations

“…The same remarks apply also to variants of Euler, such as the viscous/inviscid generalized SQG (see for instance [37,30,19,16] and references therein, and [27] for a statistical approach). We do not pursue this line of research here and plan to develop it in a future publication.…”