##
**Generalized fluid flows, their approximation and applications.**
*(English)*
Zbl 0851.76003

A fluid flow is considered as a path \(t \mapsto \xi_t\) in the group \(\mathcal D\) of volume preserving diffeomorphisms on some domain \(G \subseteq \mathbb{R}^n\). The arclength of such a path is given by \(L(\xi_t)^{t_1}_{t_0}:=\int^{t_1}_{t_0}\left(\int_G \left|{\partial i_t (x) \over \partial t}\right|^2 dx \right)^{1/2} dt\). For dimension \(n \geq 3\), not every \(\xi \in {\mathcal D}\) may be connected with the identity by a path of minimal length. To overcome this difficulty, Y. Brenier [J. Am. Math. Soc. 2, 225-255 (1989; Zbl 0697.76030)] introduced the measure-theoretic concept of generalized flow and showed that such flows with minimal action among the incompressible ones always exist.

The main theorem of this paper, which builds upon A. I. Shnirelman [Math. USSR, Sb. 56, No. 1, 79-105 (1987); translation from Mat. Sb., Nov. Ser. 128(170), No. 1, 82-109 (1985; Zbl 0725.58005)], is an approximation result saying that in dimension \(n \geq 3\) for every incompressible generalized flow there is a sequence of smooth flows with the same endpoints, and that their associated measures and actions approximate those of the generalized flow. In the second part, several applications are given; among them, estimates for the diameter of \(\mathcal D\), Hölder estimates for the metric on \(\mathcal D\), the failing of the approximation result for dimension 2, the non-existence of minimal flows on a cube, and the existence of conjugate points on geodesics in \(\mathcal D\).

The main theorem of this paper, which builds upon A. I. Shnirelman [Math. USSR, Sb. 56, No. 1, 79-105 (1987); translation from Mat. Sb., Nov. Ser. 128(170), No. 1, 82-109 (1985; Zbl 0725.58005)], is an approximation result saying that in dimension \(n \geq 3\) for every incompressible generalized flow there is a sequence of smooth flows with the same endpoints, and that their associated measures and actions approximate those of the generalized flow. In the second part, several applications are given; among them, estimates for the diameter of \(\mathcal D\), Hölder estimates for the metric on \(\mathcal D\), the failing of the approximation result for dimension 2, the non-existence of minimal flows on a cube, and the existence of conjugate points on geodesics in \(\mathcal D\).

Reviewer: A.Kriegl (Wien)

### MSC:

76A02 | Foundations of fluid mechanics |

58D05 | Groups of diffeomorphisms and homeomorphisms as manifolds |

### Keywords:

group of volume preserving diffeomorphisms; path of minimal length; measure-theoretic concept; sequence of smooth flows
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\textit{A. I. Shnirelman}, Geom. Funct. Anal. 4, No. 5, 586--620 (1994; Zbl 0851.76003)

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