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A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description. (English) Zbl 0745.76001

We consider the canonical Gibbs measure associated to an \(N\)-vortex system in a bounded domain \(\Lambda\), at inverse temperature \(\tilde\beta\) and prove that, in the limit \(N\to\infty\), \(\tilde\beta/N\to\beta\), \(\alpha N\to1\), where \(\beta\in(-8\pi,+\infty)\) (here \(\alpha\) denotes the vorticity intensity of each vortex), the one particle distribution function \(\rho^ N=\rho^ N(x)\), \(x\in\Lambda\) converges to a superposition of solutions \(\rho_ \beta\) of the mean field equation. Finally, we discuss a possible connection of the present analysis with the \(2-D\) turbulence.

MSC:

76A02 Foundations of fluid mechanics
76F99 Turbulence
82D15 Statistical mechanics of liquids
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