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Communications in Mathematical Physics

, Volume 143, Issue 3, pp 501–525 | Cite as

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

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

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) wherex0 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.

Keywords

Vortex Vorticity Equilibrium Point Bounded Domain Field Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • E. Caglioti
    • 1
  • P. L. Lions
    • 2
  • C. Marchioro
    • 3
  • M. Pulvirenti
    • 4
  1. 1.Dipartimento di FisicaUniversità di Roma “La Sapienza”RomaItaly
  2. 2.CeremadeUniversité Paris-DauphineParisFrance
  3. 3.Dipartimento di MatematicaUniversitá di Roma “La Sapienza”RomaItaly
  4. 4.Dipartimento di MatematicaUniversitá di L'AquilaItaly

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