×

zbMATH — the first resource for mathematics

The least action principle and the related concept of generalized flows for incompressible perfect fluids. (English) Zbl 0697.76030
Summary: The link between the Euler equations of perfect incompressible flows and the least action principle has been known for a long time. 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 only when the two different mappings are sufficiently close in some very strong sense [D. G. Ebin and J. Marsden in J, Ann. Math., II. Ser. 92, 102-163 (1970; Zbl 0211.574)].
In this paper, a new framework is introduced, where generalized flows are defined, in the spirit of L. C. Young [Lectures on the calculus of variations and optimal control theory (1974; Zbl 0289.49003)], 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, and an example is provided where the unique generalized solution can be explicitly computed and turns out to be genuinely probabilistic.

MSC:
76B99 Incompressible inviscid fluids
35D99 Generalized solutions to partial differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] V. Arnold, Les méthodes mathématiques de la mécanique classique, Éditions Mir, Moscow, 1976 (French). Traduit du russe par Djilali Embarek. V. I. Arnol\(^{\prime}\)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.
[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). · Zbl 0149.21704
[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 \?^\?, Deuxième édition revue et augmentée. Actualités Scientifiques et Industrielles, No. 1175, Hermann, Paris, 1965 (French). · Zbl 0136.03404
[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). · Zbl 0652.26017
[5] Ronald J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal. 88 (1985), no. 3, 223 – 270. · Zbl 0616.35055 · doi:10.1007/BF00752112 · doi.org
[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. · Zbl 0626.35059
[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. · Zbl 0211.57401 · doi:10.2307/1970699 · doi.org
[8] Ivar Ekeland and Roger Temam, Analyse convexe et problèmes variationnels, Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974 (French). Collection Études Mathématiques. 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.
[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). · Zbl 0565.60010
[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. · Zbl 0459.46001
[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. · Zbl 0536.35003
[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. · Zbl 0177.37801
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.