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Mat. Sb. (N.S.), 1985, Volume 128(170), Number 1(9), Pages 82–109 (Mi msb2019)  

This article is cited in 44 scientific papers (total in 44 papers)

On the geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid

A. I. Shnirel'man


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.

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English version:
Mathematics of the USSR-Sbornik, 1987, 56:1, 79–105

Bibliographic databases:

UDC: 514.853+517.974
MSC: 58D05, 76B99
Received: 09.03.1983 and 15.04.1985

Citation: A. I. Shnirel'man, “On the geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid”, Mat. Sb. (N.S.), 128(170):1(9) (1985), 82–109; Math. USSR-Sb., 56:1 (1987), 79–105

Citation in format AMSBIB
\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 Mat. Sb. (N.S.)
\yr 1985
\vol 128(170)
\issue 1(9)
\pages 82--109
\mathnet{http://mi.mathnet.ru/msb2019}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=805697}
\zmath{https://zbmath.org/?q=an:0725.58005}
\transl
\jour Math. USSR-Sb.
\yr 1987
\vol 56
\issue 1
\pages 79--105
\crossref{https://doi.org/10.1070/SM1987v056n01ABEH003025}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
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    6. Lorenzo Sadun, Misha Vishik, “The spectrum of the second variation of the energy for an ideal incompressible fluid”, Physics Letters A, 182:4-6 (1993), 394  crossref
    7. A. I. Shnirelman, “Generalized fluid flows, their approximation and applications”, GAFA Geom funct anal, 4:5 (1994), 586  crossref  mathscinet  zmath
    8. Ve D., “Prescribing in the Jacobian Determinant in Sobolev Spaces”, Ann. Inst. Henri Poincare-Anal. Non Lineaire, 11:3 (1994), 275–296  mathscinet  zmath  isi
    9. Chemin J., “Incompressible Perfect Fluids”, Asterisque, 1995, no. 230, 3–&  mathscinet  zmath  isi
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    17. Shnirelman A., “Weak Solutions with Decreasing Energy of Incompressible Euler Equations”, Commun. Math. Phys., 210:3 (2000), 541–603  crossref  mathscinet  zmath  adsnasa  isi
    18. Benamou J., Brenier Y., “A Computational Fluid Mechanics Solution to the Monge-Kantorovich MASS Transfer Problem”, Numer. Math., 84:3 (2000), 375–393  crossref  mathscinet  zmath  isi
    19. Felix Otto, “THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION”, Communications in Partial Differential Equations, 26:1-2 (2001), 101  crossref
    20. Khesin B., “Topology Bounds the Energy”, Introduction to the Geometry and Topology of Fluid Flows, NATO Science Series, Series II: Mathematics, Physics and Chemistry, 47, ed. Ricca R., Springer, 2001, 229–238  mathscinet  isi
    21. Benaim M., Gambaudo J., “Metric Properties of the Group of Area Preserving Diffeomorphisms”, Trans. Am. Math. Soc., 353:11 (2001), 4661–4672  crossref  mathscinet  zmath  isi
    22. Adrian Constantin, Boris Kolev, “On the geometric approach to the motion of inertial mechanical systems”, J Phys A Math Gen, 35:32 (2002), R51  crossref  mathscinet  zmath
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    24. Brenier Y. Loeper G., “A Geometric Approximation to the Euler Equations: the Vlasov-Monge-Ampere System”, Geom. Funct. Anal., 14:6 (2004), 1182–1218  crossref  mathscinet  isi
    25. A. Shnirelman, “Inverse Cascade Solutions of the Euler Equations”, Journal of Mathematical Sciences (New York), 128:2 (2005), 2818  crossref  mathscinet  zmath
    26. Jalal Shatah, Chongchun Zeng, “Geometry and a priori estimates for free boundary problems of the Euler's equation”, Comm Pure Appl Math, 61:5 (2008), 698  crossref  mathscinet  zmath
    27. Cornelia Vizman, “Geodesic Equations on Diffeomorphism Groups”, SIGMA, 4 (2008), 030, 22 pp.  mathnet  crossref  mathscinet  zmath
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    29. Shatah J., Zeng Ch., “A Priori Estimates for Fluid Interface Problems”, Commun. Pure Appl. Math., 61:6 (2008), 848–876  crossref  mathscinet  zmath  isi
    30. Ambrosio L., Figalli A., “On the Regularity of the Pressure Field of Brenier's Weak Solutions to Incompressible Euler Equations”, Calc. Var. Partial Differ. Equ., 31:4 (2008), 497–509  crossref  mathscinet  zmath  isi
    31. Khesin B., Lee P., “A Nonholonomic Moser Theorem and Optimal Transport”, J. Symplectic Geom., 7:4 (2009), 381–414  mathscinet  zmath  isi
    32. Ambrosio L., Figalli A., “Geodesics in the Space of Measure-Preserving Maps and Plans”, Arch. Ration. Mech. Anal., 194:2 (2009), 421–462  crossref  mathscinet  zmath  isi
    33. Bernot M., Figalli A., Santambrogio F., “Generalized Solutions for the Euler Equations in One and Two Dimensions”, J. Math. Pures Appl., 91:2 (2009), 137–155  crossref  mathscinet  zmath  isi
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    37. Pusateri F., “On the Limit as the Surface Tension and Density Ratio Tend to Zero for the Two-Phase Euler Equations”, J. Hyberbolic Differ. Equ., 8:2 (2011), 347–373  crossref  mathscinet  zmath  isi
    38. Lopes Filho M.C., Nussenzveig Lopes H.J., Precioso J.C., “Least Action Principle and the Incompressible Euler Equations with Variable Density”, Trans. Am. Math. Soc., 363:5 (2011), 2641–2661  crossref  mathscinet  zmath  isi
    39. Shatah J., Zeng Ch., “Local Well-Posedness for Fluid Interface Problems”, Arch. Ration. Mech. Anal., 199:2 (2011), 653–705  crossref  mathscinet  zmath  isi  elib
    40. Yann Brenier, “Remarks on the minimizing geodesic problem in inviscid incompressible fluid mechanics”, Calc. Var, 2012  crossref
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    42. Kukavica I., Tuffaha A., “Well-Posedness for the Compressible Navier–Stokes-Lame System with a Free Interface”, Nonlinearity, 25:11 (2012), 3111–3137  crossref  mathscinet  zmath  adsnasa  isi
    43. Lin Zh., Zeng Ch., “Unstable Manifolds of Euler Equations”, Commun. Pure Appl. Math., 66:11 (2013), 1803–1836  crossref  isi
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  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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