The least action principle and the related concept of generalized flows for incompressible perfect fluids

Abstract:

The link between the Euler equations of perfect incompressible flows and the Least Action Principle has been known for a long time [1]. Solutions can be considered as geodesic curves along the manifold of volume preserving mappings. Here the “shortest path problem” is investigated. Given two different volume preserving mappings at two different times, find, for the intermediate times, an incompressible flow map that minimizes the kinetic energy (or, more generally, the Action). In its classical formulation, this problem has been solved [7] only when the two different mappings are sufficiently close in some very strong sense. In this paper, a new framework is introduced, where generalized flows are defined, in the spirit of L. C. Young, as probability measures on the set of all possible trajectories in the physical space. Then the minimization problem is generalized as the “continuous linear programming” problem that is much easier to handle. The existence problem is completely solved in the case of the $d$-dimensional torus. It is also shown that under natural restrictions a classical solution to the Euler equations is the unique optimal flow in the generalized framework. Finally, a link is established with the concept of measure-valued solutions to the Euler equations [6], and an example is provided where the unique generalized solution can be explicitly computed and turns out to be genuinely probabilistic.