A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description

We consider the canonical Gibbs measure associated to a N-vortex system in a bounded domain Λ, at inverse temperature {Mathematical expression} and prove that, in the limit N→∞, {Mathematical expression}/N→β, αN→1, where β∈(-8π, + ∞) (here α denotes the vorticity intensity of each vortex), the one particle distribution function ρ{variant}N = ρ{variant}Nx, x∈Λ converges to a superposition of solutions ρ{variant}α of the following Mean Field Equation: {Mathematical expression} Moreover, we study the variational principles associated to Eq. (A.1) and prove thai, when β→-8π+, either ρ{variant}β → δx0 (weakly in the sense of measures) where x0 denotes and equilibrium point of a single point vortex in Λ, or ρ{variant}β converges to a smooth solution of (A.1) for β=-8π. Examples of both possibilities are given, although we are not able to solve the alternative for a given Λ. Finally, we discuss a possible connection of the present analysis with the 2-D turbulence. © 1992 Springer-Verlag.