Abstract:
The author studies the geometric properties of the group of volume-preserving diffeomorphisms of a region. This group is the configuration space of an ideal incompressible fluid, the trajectories of the motion of the fluid in the absence of external forces being geodesics on the group.
The author constructs configurations of the fluid in a 3-dimensional cube which cannot be connected in the group of diffeomorphisms by a trajectory of minimal length. This shows the difficulty of applying the variational method to construct nonstationary flows in the 3-dimensional case.
He shows that in the 3-dimensional case the group of diffeomorphisms has finite diameter, in contrast to the 2-dimensional case. He describes completion (as a metric space) of the group of volume-preserving diffeomorphisms of a 3-dimensional region; it consists of all measurable, not necessarily invertible volume-preserving maps of the region into itself.
Bibliography: 6 titles.
Citation:
A. I. Shnirel'man, “On the geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid”, Math. USSR-Sb., 56:1 (1987), 79–105
\Bibitem{Shn85}
\by A.~I.~Shnirel'man
\paper On the geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid
\jour Math. USSR-Sb.
\yr 1987
\vol 56
\issue 1
\pages 79--105
\mathnet{http://mi.mathnet.ru/eng/sm2019}
\crossref{https://doi.org/10.1070/SM1987v056n01ABEH003025}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=805697}
\zmath{https://zbmath.org/?q=an:0725.58005}
Linking options:
https://www.mathnet.ru/eng/sm2019
https://doi.org/10.1070/SM1987v056n01ABEH003025
https://www.mathnet.ru/eng/sm/v170/i1/p82
This publication is cited in the following 67 articles:
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Anton Izosimov, Boris Khesin, “Geometry of generalized fluid flows”, Calc. Var., 63:1 (2024)
Leandro Lichtenfelz, Gerard Misiołek, Stephen C Preston, “Axisymmetric Diffeomorphisms and Ideal Fluids on Riemannian 3-Manifolds”, International Mathematics Research Notices, 2022:1 (2022), 446
Yann Brenier, Iván Moyano, “Relaxed solutions for incompressible inviscid flows: a variational and gravitational approximation to the initial value problem”, Phil. Trans. R. Soc. A., 380:2219 (2022)
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Yann Brenier, “Various formulations and approximations of incompressible fluid motions in porous media”, Ann. Math. Québec, 46:1 (2022), 195
Dmitry Jakobson, Boris Khesin, Iosif Polterovich, “Special issue in honour of Alexander Shnirelman's 75th birthday”, Ann. Math. Québec, 46:1 (2022), 1
Martin Bauer, Cy Maor, “Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics”, Calc. Var., 60:2 (2021)
Martin Bauer, Philipp Harms, Peter W. Michor, “Fractional Sobolev metrics on spaces of immersions”, Calc. Var., 59:2 (2020)
Hannah Alpert, “Discrete configuration spaces of squares and hexagons”, J Appl. and Comput. Topology, 4:2 (2020), 263
Martin Bauer, Philipp Harms, Stephen C. Preston, “Vanishing Distance Phenomena and the Geometric Approach to SQG”, Arch Rational Mech Anal, 235:3 (2020), 1445
Aymeric Baradat, “Nonlinear instability in Vlasov type equations around rough velocity profiles”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37:3 (2020), 489
Aymeric Baradat, “Continuous dependence of the pressure field with respect to endpoints for ideal incompressible fluids”, Calc. Var., 58:1 (2019)
Luigi De Pascale, Jean Louet, Filippo Santambrogio, “The Monge problem with vanishing gradient penalization: Vortices and asymptotic profile”, Journal de Mathématiques Pures et Appliquées, 106:2 (2016), 237
Precioso J.C., “On the Regularity of the Pressure Field of Relaxed Solutions to Euler Equations with Variable Density”, J. Math. Anal. Appl., 409:1 (2014), 282–287
Lin Zh., Zeng Ch., “Unstable Manifolds of Euler Equations”, Commun. Pure Appl. Math., 66:11 (2013), 1803–1836
Yann Brenier, “Remarks on the minimizing geodesic problem in inviscid incompressible fluid mechanics”, Calc. Var, 2012
E. A. Rogozinnikov, “O vozmozhnosti postroeniya krivoi po zadannoi gruppe gomeomorfizmov”, Tr. IMM UrO RAN, 18, no. 3, 2012, 218–229
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