Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations

The three-dimensional motion of an incompressible inviscid fluid is classic ally described by the Euler equations but can also be seen, following Arnol d [1], as a geodesic on a group of volume-preserving maps. Local existence and uniqueness of minimal geodesics have been established by Ebin and Marsd en [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 tha t the limits of approximate solutions are solutions of a suitable extension of the Euler equations or, equivalently, are sharp measure-valued solution s to the Euler equations in the sense of DiPerna and Majda [14]. (C) 1999 J ohn Wiley & Sons, Inc.