Minimal Geodesics on Groups

The three-dimensional motion of an incompressible inviscid uid is classically described by the Euler equations, but can also be seen, following Arnold 1], as a geodesic on a group of volume-preserving maps. Local existence and uniqueness of minimal geodesics have been established by Ebin and Marsden 16]. In the large, for a large class of data, the existence of minimal geodesics may fail, as shown by Shnirelman 26]. For such data, we show that the limits of approximate solutions are solutions of a suitable extension of the Euler equations or, equivalently, as sharp measure-valued solutions to the Euler equations in the sense of DiPerna and Majda 14].