A SPECIAL-CLASS OF STATIONARY FLOWS FOR 2-DIMENSIONAL EULER EQUATIONS - A STATISTICAL-MECHANICS DESCRIPTION

We consider the canonical Gibbs measure associated to a N-vortex system in a bounded domain LAMBDA, at inverse temperature beta-approximately and prove that, in the limit N --> infinity, beta-approximately/N --> beta, alpha-N --> 1, where beta epsilon (- 8-pi, + infinity) (here alpha denotes the vorticity intensity of each vortex), the one particle distribution function rho(N) = rho(N)(x), x is-an-element-of LAMBDA converges to a superposition of solutions rho(beta) of the following Mean Field Equation: [GRAPHICS] Moreover, we study the variational principles associated to Eq. (A.1) and prove that, when beta --> - 8-pi+, either rho(beta) --> delta(x0) (weakly in the sense of measures) where x0 denotes an equilibrium point of a single point vortex in LAMBDA, or rho(beta) converges to a smooth solution of (A.1) for beta = - 8-pi. Examples of both possibilities are given, although we are not able to solve the alternative for a given LAMBDA. Finally, we discuss a possible connection of the present analysis with the 2-D turbulence.