Geodesics in the space of measure-preserving maps and plans

Details Geodesics in the space of measure-preserving maps and plans Geodesics in the space of measure-preserving maps and plans

2007-01-29

arxiv.org/abs/math/0701848v3

Mathematics

Analysis of PDEs

Scientific paper

10.1007/s00205-008-0189-2

We study Brenier's variational models for incompressible Euler equations. These models give rise to a relaxation of the Arnold distance in the space of measure-preserving maps and, more generally, measure-preserving plans. We analyze the properties of the relaxed distance, we show a close link between the Lagrangian and the Eulerian model, and we derive necessary and sufficient optimality conditions for minimizers. These conditions take into account a modified Lagrangian induced by the pressure field. Moreover, adapting some ideas of Shnirelman, we show that, even for non-deterministic final conditions, generalized flows can be approximated in energy by flows associated to measure-preserving maps.

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