Calculus of Variations and Geometric Measure Theory

L. Ambrosio - A. Figalli

Geodesics in the space of measure-preserving maps and plans

created by ambrosio on 29 Jan 2007
modified by figalli on 08 Nov 2008


Accepted Paper

Inserted: 29 jan 2007
Last Updated: 8 nov 2008

Journal: Arch. Rat. Mech. Anal.
Year: 2007


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.

Keywords: relaxation, optimal transportation, Incompressible Euler equations, Arnold distance