, Volume 143, Issue 3, pp 501-525

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

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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 ϱN = 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 ϱβ → δ x 0 (weakly in the sense of measures) wherex 0 denotes and equilibrium point of a single point vortex in Λ, or ϱβ 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.

Work partially supported by CNR (GNFM) and MPI
Communicated by J. L. Lebowitz