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.

Point vortices for inviscid generalized surface quasi-geostrophic models

“…See for instance [CCC + 12, KRYZ16, CGSI17] for relevant results. Finally we remark that the validation of the point vortex motion proved here bolsters the statistical mechanics of point vortices discussed in [GR18], where the authors extend results on Euler equations from [CLMP92, CLMP95, Lio98, BG99].…”

Energy conditional measures and 2D turbulence

“…centered Bernoulli random variables, Benfatto et al [3] proved that the canonical Gibbs measures of the point vortices, with appropriately regularized Green functions, converge to the Gaussian measure µ β,γ (dω) = e −βH−γE dω (β, γ > 0, H and E are the energy and enstrophy functionals), which are invariant for the 2D Euler flow. In the recent work [14], analogous result was proved without smoothing the Green function; see [12] for related result concerning the generalised inviscid surface quasi-geostropic equations.…”

Regularized vortex approximation for 2D Euler equations with transport noise

“…The two terms in ( 27) can be estimated using the boundedness of the coefficients, Burkholder-Davis-Gundy inequality for the second one, Hölder inequality and (17) to obtain…”

“…N (0, N ), then the vortex system converges a.s. to a random, stationary solution to the Euler equations with one-time marginals distributed as µ; the result is generalized also to solutions whose one-time marginals are absolutely continuous with respect to µ. For other convergence results in the deterministic case the reader can refer to [17,15,20].…”