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.


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


Zbl 0745.76001
Full Text: DOI


[1] [CK] Chanillo, S., Kiessling, M.H.K.: Commun. Math. Phys.160, 217–238 (1994) · Zbl 0821.35044
[2] [CLMP] Caglioti, E., Lions, P.L., Marchioro, C., Pulvirenti, M.: Commun. Math. Phys.143, 501–525 (1992) · Zbl 0745.76001
[3] [ES] Eyink, G.L., Spohn, H.: J. Stat. Phys.70, Nov. 3/4 (1993)
[4] [GL] Gogny, D., Lions, P.L.: RAIRO Modél. Math. et Anal. Num.23, 137–153 (1989)
[5] [GNN] Gidas, B., Ni, W.M., Nirenberg, L.: Commun. Math. Phys.68, 203–243 (1979) · Zbl 0425.35020
[6] [H] Hopf, E.: J. Rat. Mech. Anal.1, 87–123 (1952)
[7] [Ki] Kiessling, M.K.H.: Comm. Pure Appl. Math.46, 27–56 (1993) · Zbl 0811.76002
[8] [Ki]2 Kiessling, M.K.H.: J. Stat. Phys.55, 203–257 (1989)
[9] [LS] Leray, J., Schauder: Topology et equations functionnels, Ann. Sci. Ecole Norm. Sup.3, 45–78 (1934)
[10] [MS] Messer, J., Spohn, H.: J. Stat. Phys.29, 561–578 (1982)
[11] [M] Moser, J.: Indiana Univ. Math. J.20, 1077–1092 (1971) · Zbl 0213.13001
[12] [MMS] Montgomery, D., Matthaeus, W.H., Stribling, W.T., Martinez, D., Oughton, S.: Phys. FluidsA4, 3–6 (1992); · Zbl 0850.76485
[13] Matthaeus, W.H., Stribling, W.T., Martinez, D., Oughton, S., Montgomery, D.: Phys. Rev. Lett.66, 2731 (1991)
[14] [Mo] Montgomery, D.: Phys. Lett.39A, 7–8 (1972)
[15] [MoJ] Montgomery, D., Joyce, G.: Phys. Fluids17, 1139–1145 (1974)
[16] [MP] Marchioro, C., Pulvirenti, M.: Mathematical Theory of Incompressible Nonviscous Fluids, Appl. Math. Sci.96, Berlin, Heidelberg, New York: Springer 1994 · Zbl 0789.76002
[17] [NaSu] Nagasaki, K., Suzuki, T.: Asymptotic Analysis3, 173–188 (1990)
[18] [O] Onsager, L.: Suppl. Nuovo Cimento279 (1949)
[19] [Po] Pohozaev, S.I.: Soviet Math. Dokl.6, 1408–1411 (1965)
[20] [SON] Smith, R.A., O’Neil, T.: Phys. Fluids.B2, 2961–2975 (1990)
[21] [Su] Suzuki, T.: Ann. Inst. H. Poincaré9, 4, 367–398 (1992)
[22] [SuNa] Suzuki, T., Nagasaki, K.: Trans. Am. Math. Soc.309 (1988)
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