Alexander Shnirelman - Topics in Mathematical Fluid Dynamics / Part 5

The Ideal Incompressible Fluid is the most fundamental model of a continuous media. In this model, the configuration space of the fluid is the group D of volume- preserving diffeomorphisms of the flow domain M. D is an infinite-dimensonal manifold endowed with a weak Riemannian metric (the kinetic energy). The fluid flows (in the absence of the external forces) are the geodesics in this metric. There are two basic problems about the geodesics:
Find the geodesic trajectory for the given initial fluid configuration and velocity (the initial-value problem);
Find the geodesic for given fluid configurations at two different time moments, say at t=0 and t=1 (the 2-point problem).