×

zbMATH — the first resource for mathematics

A maximum-entropy principle for two-dimensional perfect fluid dynamics. (English) Zbl 0935.76530
Summary: We use Kullback entropy for Young measures to define statistical equilibrium states for a two-dimensional incompressible flow of a perfect fluid. This approach is justified, as it gives a concentration property about the equilibrium state in the phase space. It might give a statistical understanding of the appearance of coherent structures in two-dimensional turbulence.

MSC:
76F99 Turbulence
76M35 Stochastic analysis applied to problems in fluid mechanics
35Q35 PDEs in connection with fluid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. Albeverio and A. Cruzerio, Global flows with invariant Gibbs measures for Euler and Navier-Stokes two-dimensional fluids,Commun. Math. Phys. 129:431-444 (1990). · Zbl 0702.76041 · doi:10.1007/BF02097100
[2] S. Albeverio, M. Ribeiro de Faria, and R. Hoegh Krohn, Stationary measures for the periodic Euler flow in two dimensions,J. Stat. Phys. 20:585-595 (1979). · doi:10.1007/BF01009512
[3] V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits,Ann. Inst. Fourier 16:319-361 (1966).
[4] V. I. Arnold, On ana priori estimate in the theory of hydrodynamical stability,Am. Math. Soc. Transl. 79:267-269 (1969).
[5] R. Azencott, Grandes déviations et applications, inEcole d’été de Probabilités de Saint-Flour, VIII Lecture Notes in Mathematics, No. 774 (Springer, Berlin, 1978).
[6] P. Baldi, Large deviations and stochastic homogenization,Ann. Mat. Pura Appl. 4(151):161-177 (1988). · Zbl 0654.60024 · doi:10.1007/BF01762793
[7] C. Bardos, Existence et unicité de la solution de l’équation d’Euler en dimension deux,J. Math. Anal. Appl. 40:769-790 (1972). · Zbl 0249.35070 · doi:10.1016/0022-247X(72)90019-4
[8] C. Boldrighini and S. Frigio, Equilibrium states for a plane incompressible perfect fluid,Commun. Math. Phys. 72:55-76 (1980). · Zbl 0453.76019 · doi:10.1007/BF01200111
[9] B. Castaing, Conséquences d’un principe d’extremum en turbulence,J. Phys. (Paris)50:147-156 (1989).
[10] G. H. Cottet, Analyse numérique des méthodes particulaires pour certains problèmes non linéaires, Thèse, Université Paris VI (1987).
[11] G. S. Deem and N. J. Zabusky, Vortex waves: Stationary V-states, interactions, recurrence, and breaking,Phys. Rev. Lett. 40:859-862 (1978). · doi:10.1103/PhysRevLett.40.859
[12] T. Dumont and M. Schatzman, to appear.
[13] R. S. Ellis,Entropy, Large Deviations and Statistical Mechanics (Springer-Verlag, 1985). · Zbl 0566.60097
[14] J. Fröhlich and D. Ruelle, Statistical mechanics of vortices in an inviscid two dimensional fluid,Commun. Math. Phys. 87:1-36 (1982). · Zbl 0505.76037 · doi:10.1007/BF01211054
[15] E. J. Hopfinger, Turbulence and vortices in rotating fluids, inProceedings of the XVIIth International Congress of Theoretical and Applied Mechanics (North-Holland, Amsterdam, 1989).
[16] E. T. Jaynes, Where do we go from here? inMaximum Entropy and Bayesian Methods in Inverse Problems, C. Ray Smith and W. T. Grandy, eds. (Reidel, 1985).
[17] M. Jirina, On regular conditional probabilities,Czech. Math. J. 9:445 (1959).
[18] T. Kato, On the classical solutions of the two-dimensional non stationary Euler equation,Arch. Rat. Mech. Anal. 25:302-324 (1967). · Zbl 0166.45302 · doi:10.1007/BF00251588
[19] R. H. Kraichnan, Statistical dynamics of two-dimensional flow,J. Fluid Mech. 67:155-175 (1975). · Zbl 0297.76042 · doi:10.1017/S0022112075000225
[20] R. H. Kraichnan and D. Montgomery,Rep. Prog. Phys. 43:547 (1980). · doi:10.1088/0034-4885/43/5/001
[21] T. D. Lee,Q. Appl. Math. 10:69 (1952).
[22] C. E. Leith, Minimum enstrophy vortices,Phys. Fluids 27:1388-1395 (1984). · Zbl 0572.76046 · doi:10.1063/1.864781
[23] J. Michel and R. Robert, To appear.
[24] J. Miller, Statistical mechanics of Euler equations in two dimensions,Phys. Rev. Lett. 65:2137-2140 (1990). · Zbl 1050.82553 · doi:10.1103/PhysRevLett.65.2137
[25] D. Montgomery, Maximal entropy in fluid and plasma turbulence, inMaximum Entropy and Bayesian Methods in Inverse Problems, C. Ray Smith and W. T. Grandy, eds. (Reidel, 1985). · Zbl 0574.76127
[26] J. M. Nguyen Duc and J. Sommeria, Experimental characterization of steady two-dimensional vortex couples,J. Fluid Mech. 192:175-192 (1988). · doi:10.1017/S002211208800182X
[27] E. A. Novikov, Dynamics and statistics of a system of vortices,Sov. Phys. JETP 41:937-943 (1976).
[28] P. J. Olver,Applications of Lie Groups to Differential Equations (Springer-Verlag, 1986). · Zbl 0588.22001
[29] L. Onsager, Statistical hydrodynamics,Nuovo Cimento Suppl. 6:279 (1949). · doi:10.1007/BF02780991
[30] E. A. Overman and N. J. Zabusky, Evolution and merging of isolated vortex structures,Phys. Fluids 25:1297 (1982). · Zbl 0489.76033 · doi:10.1063/1.863907
[31] Y. B. Poitin and T. S. Lundgren, Statistical mechanics of two dimensional vortices in a bounded container,Phys. Fluids 10:1459-1470 (1976). · Zbl 0339.76013 · doi:10.1063/1.861347
[32] R. Robert, Concentration et entropie pour les mesures d’Young,C. R. Acad. Sci. Paris Ser. I 309:757-760 (1989). · Zbl 0709.94010
[33] R. Robert, Concentration and entropy for Young measures, Preprint, Laboratoire d’analyse numérique de Lyon, no. 85 (1989).
[34] R. Robert, Etats d’équilibre statistique pour l’écoulement bidimensionnel d’un fluide parfait,C. R. Acad. Sci. Paris Ser. I 311:575-578 (1990). · Zbl 0707.76002
[35] R. Robert and J. Sommeria, Statistical equilibrium states for two-dimensional flows,J. Fluid Mech., to appear. · Zbl 0850.76025
[36] P. G. Saffman and G. R. Baker, Vortex interactions,Annu. Rev. Fluid Mech. 11:95-122 (1979). · Zbl 0434.76001 · doi:10.1146/annurev.fl.11.010179.000523
[37] D. Serre, Les invariants du premier ordre de l’équation d’Euler en dimension trois,Physica 13D:105-136 (1984). · Zbl 0586.76025
[38] J. Sommeria and M. A. Denoix, to appear.
[39] J. Sommeria, S. D. Meyers, and H. L. Swinney, Laboratory simulation of Jupiter’s Great Red Spot,Nature 331:1 (1988). · doi:10.1038/331689a0
[40] J. Sommeria, C. Nore, T. Dumont, and R. Robert, Théorie statistique de la tache rouge de Jupiter,C. R. Acad. Sci. Paris Ser. II 312:999-1005 (1991).
[41] J. Sommeria, C. Staquet, and R. Robert, Final equilibrium state of a two-dimensional shear layer,J. Fluid Mech., to appear. · Zbl 0738.76031
[42] V. I. Youdovitch, Non stationary flow of an ideal incompressible liquid,Zh. Vych. Mat. 3:1032-1066 (1963).
[43] L. C. Young, Generalized surfaces in the calculus of variations,Ann. Math. 43:84-103 (1942). · Zbl 0063.09081 · doi:10.2307/1968882
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.