Communications in Mathematical Physics

, Volume 72, Issue 1, pp 55–76 | Cite as

Equilibrium states for a plane incompressible perfect fluid

  • Carlo Boldrighini
  • Sandro Frigio


We associate to the plane incompressible Euler equation with periodic conditions the corresponding Hopf equation, as an equation for measures on the space of solenoidal distributions. We define equilibrium states as the solutions of the stationary Hopf equation. We find a class of equilibrium states which corresponds to a class of infinitely divisible distributions, and investigate the properties of gaussian and poissonian states. Equilibrium dynamics for a class of poissonian states is constructed by means of the Onsager vortex equations.


Vortex Neural Network Statistical Physic Equilibrium State Complex System 
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  1. 1.
    Lee, T.D.: Quart. Appl. Math.10, 69 (1952)Google Scholar
  2. 2.
    Novikov, E.A.: Sov. Phys. JETP41, 937 (1976)Google Scholar
  3. 3.
    Gurievich, B.M., Suhov, Ju.M.: Commun. Math. Phys.49, 63 (1976)Google Scholar
  4. 4.
    Dobrushin, R.L., Suhov, Ju.M.: Lecture notes in physics, Vol. 80, p. 325. Berlin, Heidelberg, New York: Springer 1978Google Scholar
  5. 5.
    Fox, D.G., Orszag, S.A.: Phys. Rev. Lett.16, 169 (1973)Google Scholar
  6. 6.
    Lanford, O.E. III, Commun. Math. Phys.9, 169 (1968);11, 257 (1969)Google Scholar
  7. 7.
    Fritz, J., Dobrushin, R.L.: Commun. Math. Phys.57, 67 (1977)Google Scholar
  8. 8.
    Lanford, O.E. III: Lecture notes in physics, Vol. 38, p. 1. Berlin, Heidelberg, New York: Springer 1975Google Scholar
  9. 9.
    Kato, T.: Arch. Rat. Anal.25, 188 (1967)Google Scholar
  10. 10.
    Bardos, C.: I. Math. An. Appl.40, 769 (1972)Google Scholar
  11. 11.
    Boldrighini, C.: Introduzione alla Fluidodinamica, Quaderni del C.N.R., Roma., 1979Google Scholar
  12. 12.
    Lee, J.: Phys. Fluids20, 1250 (1977)Google Scholar
  13. 13.
    Gelfand, I.M., Vilenkin, M.Y.: Generalized functions, Vol. IV. New York: Academic Press 1964Google Scholar
  14. 14.
    Reed, M., Simon, B.: Functional analysis, Vol. 1. New York: Academic Press 1972Google Scholar
  15. 15.
    Gnedenko, B.V., Kolmogorov, A.M.: Limit distributions for sums of independent random variables. Cambridge: Addison Wesley 1954Google Scholar
  16. 16.
    Skorokhod, A.V.: Stochastic processes with independent increments. Moscow: Nauka 1964Google Scholar
  17. 17.
    Novikov, E.A.: Teoretičeskaja i Matematičeskaja Fizika. To appearGoogle Scholar
  18. 18.
    Albeverio, S., Ribeiro de Faria, M., Høegh-Krohn, R.: Centre de Physique Teorique, CNRS Marseille Université d'Aix, Marseille II, Uer Scientifique di Luminy, preprint, 1978Google Scholar
  19. 19.
    Goldstein, S., Lebowitz, J.L., Aizenman, M.: Lecture notes in physics, Vol. 38, pp. 112. Berlin, Heidelberg, New York: Springer 1975Google Scholar
  20. 20.
    Sinai, Ya.G.: Private communicationGoogle Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Carlo Boldrighini
    • 1
  • Sandro Frigio
    • 1
    • 2
  1. 1.Istituto MatematicoUniversità di CamerinoCamerinoItaly
  2. 2.C.N.R. fellowship, G.N.F.M.Italy

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