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A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description. II. (English) Zbl 0840.76002

Summary: We continue and conclude our analysis started in part I [ibid. 143, No. 3, 501-525 (1992; Zbl 0745.76001)] by discussing the microcanonical Gibbs measure associated to a \(N\)-vortex system in a bounded domain. We investigate the mean-field limit for such a system and study the corresponding microcanonical variational principle for the mean-field equation. We discuss and achieve the equivalence of the ensembles for domains in which we have the concentration at \(\beta\to (-8\pi)^+\) in the canonical framework. In this case we have the uniqueness of the solutions of the mean-field equation. For the other kind of domains, for large values of the energy, there is no equivalence, the entropy is not a concave function of the energy, and the mean-field equation has more than one solution. In both situations, we have concentration when the energy diverges. The microcanonical mean field limit for the \(N\)-vortex system is proven in the case of equivalence of ensembles.

MSC:

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

Citations:

Zbl 0745.76001
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References:

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