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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
\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
\jour Math. USSR-Sb.
\yr 1987
\vol 56
\issue 1
\pages 79--105

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  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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