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The least action principle and the related concept of generalized flows for incompressible perfect fluids

Author: Yann Brenier
Journal: J. Amer. Math. Soc. 2 (1989), 225-255
MSC: Primary 58D05; Secondary 35Q10, 49H05, 58E30, 58F11, 76A02, 76C05
MathSciNet review: 969419
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Abstract: The link between the Euler equations of perfect incompressible flows and the Least Action Principle has been known for a long time [1]. Solutions can be considered as geodesic curves along the manifold of volume preserving mappings. Here the ``shortest path problem'' is investigated. Given two different volume preserving mappings at two different times, find, for the intermediate times, an incompressible flow map that minimizes the kinetic energy (or, more generally, the Action). In its classical formulation, this problem has been solved [7] only when the two different mappings are sufficiently close in some very strong sense. In this paper, a new framework is introduced, where generalized flows are defined, in the spirit of L. C. Young, as probability measures on the set of all possible trajectories in the physical space. Then the minimization problem is generalized as the ``continuous linear programming'' problem that is much easier to handle. The existence problem is completely solved in the case of the $ d$-dimensional torus. It is also shown that under natural restrictions a classical solution to the Euler equations is the unique optimal flow in the generalized framework. Finally, a link is established with the concept of measure-valued solutions to the Euler equations [6], and an example is provided where the unique generalized solution can be explicitly computed and turns out to be genuinely probabilistic.

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  • [1] V. Arnold, Les méthodes mathématiques de la mécanique classique, Éditions Mir, Moscow, 1976 (French). Traduit du russe par Djilali Embarek. MR 0474391
    V. I. Arnol′d, Mathematical methods of classical mechanics, Springer-Verlag, New York-Heidelberg, 1978. Translated from the Russian by K. Vogtmann and A. Weinstein; Graduate Texts in Mathematics, 60. MR 0690288
  • [2] V. I. Arnold and A. Avez, Problèmes ergodiques de la mécanique classique, Monographies Internationales de Mathématiques Modernes, No. 9, Gauthier-Villars, Éditeur, Paris, 1967 (French). MR 0209436
  • [3] N. Bourbaki, Éléments de mathématique. Fasc. XIII. Livre VI: Intégration. Chapitres 1, 2, 3 et 4: Inégalités de convexité, Espaces de Riesz, Mesures sur les espaces localement compacts, Prolongement d’une mesure, Espaces 𝐿^{𝑝}, Deuxième édition revue et augmentée. Actualités Scientifiques et Industrielles, No. 1175, Hermann, Paris, 1965 (French). MR 0219684
  • [4] Yann Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 19, 805–808 (French, with English summary). MR 923203
  • [5] Ronald J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal. 88 (1985), no. 3, 223–270. MR 775191,
  • [6] Ronald J. DiPerna and Andrew J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys. 108 (1987), no. 4, 667–689. MR 877643
  • [7] David G. Ebin and Jerrold Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid., Ann. of Math. (2) 92 (1970), 102–163. MR 0271984,
  • [8] Ivar Ekeland and Roger Temam, Analyse convexe et problèmes variationnels, Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974 (French). Collection Études Mathématiques. MR 0463993
    Ivar Ekeland and Roger Temam, Convex analysis and variational problems, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. Translated from the French; Studies in Mathematics and its Applications, Vol. 1. MR 0463994
  • [9] S. T. Rachev, The Monge-Kantorovich problem on mass transfer and its applications in stochastics, Teor. Veroyatnost. i Primenen. 29 (1984), no. 4, 625–653 (Russian). MR 773434
  • [10] Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
  • [11] Luc Tartar, The compensated compactness method applied to systems of conservation laws, Systems of nonlinear partial differential equations (Oxford, 1982) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 111, Reidel, Dordrecht, 1983, pp. 263–285. MR 725524
  • [12] L. C. Young, Lectures on the calculus of variations and optimal control theory, Foreword by Wendell H. Fleming, W. B. Saunders Co., Philadelphia-London-Toronto, Ont., 1969. MR 0259704

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Additional Information

Keywords: Euler equations, incompressible flows, continuous linear programming, path integrals, least action principle
Article copyright: © Copyright 1989 American Mathematical Society