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

Abstract

We consider the canonical Gibbs measure associated to aN-vortex system in a bounded domain Λ, at inverse temperature$\widetilde\beta$ and prove that, in the limitN→∞,$\widetilde\beta$/N→β, αN→1, where β∈(−8π, + ∞) (here α denotes the vorticity intensity of each vortex), the one particle distribution function ϱ^N = ϱ^N x,x∈Λ converges to a superposition of solutions ϱ _α of the following Mean Field Equation:

$$\left\{ {\begin{array}{*{20}c} {\varrho _{\beta (x) = } \frac{{e^{ - \beta \psi } }}{{\mathop \smallint \limits_\Lambda e^{ - \beta \psi } }}; - \Delta \psi = \varrho _\beta in\Lambda } \\ {\psi |_{\partial \Lambda } = 0.} \\ \end{array} } \right.$$

Moreover, we study the variational principles associated to Eq. (A.1) and prove thai, when β→−8π⁺, either ϱ_β → δ _{x 0} (weakly in the sense of measures) wherex ₀ denotes and equilibrium point of a single point vortex in Λ, or ϱ_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.