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Generalized fluid flows, their approximation and applications. (English) Zbl 0851.76003
A fluid flow is considered as a path $$t \mapsto \xi_t$$ in the group $$\mathcal D$$ of volume preserving diffeomorphisms on some domain $$G \subseteq \mathbb{R}^n$$. The arclength of such a path is given by $$L(\xi_t)^{t_1}_{t_0}:=\int^{t_1}_{t_0}\left(\int_G \left|{\partial i_t (x) \over \partial t}\right|^2 dx \right)^{1/2} dt$$. For dimension $$n \geq 3$$, not every $$\xi \in {\mathcal D}$$ may be connected with the identity by a path of minimal length. To overcome this difficulty, Y. Brenier [J. Am. Math. Soc. 2, 225-255 (1989; Zbl 0697.76030)] introduced the measure-theoretic concept of generalized flow and showed that such flows with minimal action among the incompressible ones always exist.
The main theorem of this paper, which builds upon A. I. Shnirelman [Math. USSR, Sb. 56, No. 1, 79-105 (1987); translation from Mat. Sb., Nov. Ser. 128(170), No. 1, 82-109 (1985; Zbl 0725.58005)], is an approximation result saying that in dimension $$n \geq 3$$ for every incompressible generalized flow there is a sequence of smooth flows with the same endpoints, and that their associated measures and actions approximate those of the generalized flow. In the second part, several applications are given; among them, estimates for the diameter of $$\mathcal D$$, Hölder estimates for the metric on $$\mathcal D$$, the failing of the approximation result for dimension 2, the non-existence of minimal flows on a cube, and the existence of conjugate points on geodesics in $$\mathcal D$$.
Reviewer: A.Kriegl (Wien)

##### MSC:
 76A02 Foundations of fluid mechanics 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
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##### References:
 [1] S. Alperin, Approximation to and by measure preserving homeomorphisms, J. of London Math. Soc. 18 (1978), 305–315. · Zbl 0405.28014 [2] V.I. Arnold, Sur la geometrie differentielle des groupes de Lie de dimension infinie et ces applications a l’hydrodynamiques des fluid parfaits, Ann. Inst. Fourier, Grenoble 16 (1966), 319–361. [3] A.M. Bloch, H. Flashka, T. Ratiu, A Schur-Horn-Kostant convexity theory for the diffeomorphism group of the annulus, Inventionnes Mathematica 113:3 (1993), 511–530. · Zbl 0806.22012 [4] Y. Brenier, The Least Action Principle and the related concept of generalized flows for incompressible perfect fluids, Journal of the American Mathematical Society 2:2 (1989). · Zbl 0697.76030 [5] Y. Brenier, A dual Least Action Principle for the motion of an ideal incompressible fluid, preprint November 1991, 1–30. [6] D.G. Ebin, J. Mardsen, Groups of diffemorphisms and the motion of an ideal incompressible fluid, Ann. of Math. 92:2 (1970), 102–163. · Zbl 0211.57401 [7] Y. Eliashberg, T. Ratiu, The diameter of the symplectomorphism group is infinite, Invent. Math. 103 (1990), 327–370. · Zbl 0725.58006 [8] N. Grossman, Hilbert manifolds without epiconjugate points, Proc. Amer. Math. Soc. 16 (1965), 1365–1371. · Zbl 0135.40204 [9] G. Misiolek, Conjugate points inD {$$\mu$$}(Ti), Preprint, November 1993. [10] J. Moser, On the volume element on a manifold. Trans. Amer. Math. Soc. 120 (1965), 286–294. · Zbl 0141.19407 [11] J. Moser, Remark on the smooth approximation of volume-preserving home-omorphisms by symplectic diffeomorphisms, Zürich Forschungsinst. für Mathematik ETH, preprint, October 1992. [12] Yu.A. Neretin, Categories of bistochastic measures, and representations of some infinite-dimensional groups, Russian Acad. Sci. Sb. Math. 75:1 (1993), 197–219. · Zbl 0774.58006 [13] A.I. Shnirelman, On the geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid, Math. USSR Sbornik 56:1 (1987), 79–105, 1–30. · Zbl 0725.58005 [14] A.I. Shnirelman, Attainable diffeomorphisms, Geometric And Functional Analysis 3:3 (1993), 279–294. · Zbl 0784.57018 [15] A.I. Shnirelman, Lattice theory and the flows of ideal incompressible fluid, Russian Journal of Mathematical Physics 1:1 (1993), 105–114. · Zbl 0874.35096 [16] A.M. Vershik, Multivalued mappings with invariant measure and the Markov operators (in Russian), Zapiski Nauchnykh Seminarov LOMI, Leningrad 72 (1977), 121–160. · Zbl 0408.28014 [17] L.C. Young, Lectures on the calculus of variations and optimal control theory, Chelsea, 1980, New York.
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