Communications in Mathematical Physics

, Volume 174, Issue 2, pp 229–260 | Cite as

A special class of stationary flows for two-dimensional euler equations: A statistical mechanics description. Part II

  • E. Caglioti
  • P. L. Lions
  • C. Marchioro
  • M. Pulvirenti
Article

Abstract

We continue and conclude our analysis started in Part I (see [CLMP]) 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 Microcanonnical 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 β→(−8π)+ 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.

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Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • E. Caglioti
    • 1
  • P. L. Lions
    • 2
  • C. Marchioro
    • 1
  • M. Pulvirenti
    • 1
  1. 1.Dipartimento di MatematicaUniversità di Roma “La Sapienza”RomaItaly
  2. 2.CeremadeUniversité Paris-DauphineParisFrance

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