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

A. I. Shnirel'man

Sbornik. Mathematics, Tome 56 (1987) no. 1, pp. 79-105

Résumé

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.