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

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

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:
1. V. I. Arnol'd, “First steps in symplectic topology”, Russian Math. Surveys, 41:6 (1986), 1–21
2. O. V. Troshkin, “Topological analysis of the structure of hydrodynamic flows”, Russian Math. Surveys, 43:4 (1988), 153–190
3. Eliashberg Y., Ratiu T., “The Diameter of the Symplectomorphism Group Is Infinite”, Invent. Math., 103:2 (1991), 327–340
4. Y. Brenier, “The dual Least Action Problem for an ideal, incompressible fluid”, Arch Rational Mech Anal, 122:4 (1993), 323
5. A. I. Shnirelman, “Attainable diffeomorphisms”, GAFA Geom funct anal, 3:3 (1993), 279
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
7. A. I. Shnirelman, “Generalized fluid flows, their approximation and applications”, GAFA Geom funct anal, 4:5 (1994), 586
8. Ve D., “Prescribing in the Jacobian Determinant in Sobolev Spaces”, Ann. Inst. Henri Poincare-Anal. Non Lineaire, 11:3 (1994), 275–296
9. Chemin J., “Incompressible Perfect Fluids”, Asterisque, 1995, no. 230, 3–&
10. Brenier Y., “A Homogenized Model for Vortex Sheets”, Arch. Ration. Mech. Anal., 138:4 (1997), 319–353
11. Laurent Younes, “Computable Elastic Distances Between Shapes”, SIAM J Appl Math, 58:2 (1998), 565
12. Shkoller S., “Geometry and Curvature of Diffeomorphism Groups with H-1 Metric and Mean Hydrodynamics”, J. Funct. Anal., 160:1 (1998), 337–365
13. Otto F., “Evolution of Microstructure in Unstable Porous Media Flow: a Relaxational Approach”, Commun. Pure Appl. Math., 52:7 (1999), 873–915
14. Brenier Y., “Minimal Geodesics on Groups of Volume-Preserving Maps and Generalized Solutions of the Euler Equations”, Commun. Pure Appl. Math., 52:4 (1999), 411–452
15. Shkoller S., “Analysis on Groups of Diffeomorphisms of Manifolds with Boundary and the Averaged Motion of a Fluid”, J. Differ. Geom., 55:1 (2000), 145–191
16. Gambaudo J., Lagrange M., “Topological Lower Bounds on the Distance Between Area Preserving Diffeoomorphisms”, Bol. Soc. Bras. Mat., 31:1 (2000), 9–27
17. Shnirelman A., “Weak Solutions with Decreasing Energy of Incompressible Euler Equations”, Commun. Math. Phys., 210:3 (2000), 541–603
18. Benamou J., Brenier Y., “A Computational Fluid Mechanics Solution to the Monge-Kantorovich MASS Transfer Problem”, Numer. Math., 84:3 (2000), 375–393
19. Felix Otto, “THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION”, Communications in Partial Differential Equations, 26:1-2 (2001), 101
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
21. Benaim M., Gambaudo J., “Metric Properties of the Group of Area Preserving Diffeomorphisms”, Trans. Am. Math. Soc., 353:11 (2001), 4661–4672
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
23. Brenier Y., Gangbo W., “L-P Approximation of Maps by Diffeomorphisms”, Calc. Var. Partial Differ. Equ., 16:2 (2003), 147–164
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
25. A. Shnirelman, “Inverse Cascade Solutions of the Euler Equations”, Journal of Mathematical Sciences (New York), 128:2 (2005), 2818
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
27. Cornelia Vizman, “Geodesic Equations on Diffeomorphism Groups”, SIGMA, 4 (2008), 030, 22 pp.
28. Yann Brenier, “Generalized solutions and hydrostatic approximation of the Euler equations”, Physica D: Nonlinear Phenomena, 237:14-17 (2008), 1982
29. Shatah J., Zeng Ch., “A Priori Estimates for Fluid Interface Problems”, Commun. Pure Appl. Math., 61:6 (2008), 848–876
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
31. Khesin B., Lee P., “A Nonholonomic Moser Theorem and Optimal Transport”, J. Symplectic Geom., 7:4 (2009), 381–414
32. Ambrosio L., Figalli A., “Geodesics in the Space of Measure-Preserving Maps and Plans”, Arch. Ration. Mech. Anal., 194:2 (2009), 421–462
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
34. Ambrosio L., “Variational Models for Incompressible Euler Equations”, Discrete Contin. Dyn. Syst.-Ser. B, 11:1 (2009), 1–10
35. Brenier Ya., “Hidden Convexity in Some Nonlinear PDEs From Geomety and Physics”, J. Convex Anal., 17:3-4, SI (2010), 945–959
36. Figalli A., Mandorino V., “Fine Properties of Minimizers of Mechanical Lagrangians with Sobolev Potentials”, Discret. Contin. Dyn. Syst., 31:4, SI (2011), 1325–1346
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
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
39. Shatah J., Zeng Ch., “Local Well-Posedness for Fluid Interface Problems”, Arch. Ration. Mech. Anal., 199:2 (2011), 653–705
40. Yann Brenier, “Remarks on the minimizing geodesic problem in inviscid incompressible fluid mechanics”, Calc. Var, 2012
41. E. A. Rogozinnikov, “O vozmozhnosti postroeniya krivoi po zadannoi gruppe gomeomorfizmov”, Tr. IMM UrO RAN, 18, no. 3, 2012, 218–229
42. Kukavica I., Tuffaha A., “Well-Posedness for the Compressible Navier–Stokes-Lame System with a Free Interface”, Nonlinearity, 25:11 (2012), 3111–3137
43. Lin Zh., Zeng Ch., “Unstable Manifolds of Euler Equations”, Commun. Pure Appl. Math., 66:11 (2013), 1803–1836
44. 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
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