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 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.

Continuity equations and ODE flows with non-smooth velocity

“…The viewpoint adopted in this section is very close in spirit to Young's theory [106] of generalized surfaces and controls (a theory that also has remarkable applications to nonlinear PDEs [71,99] and calculus of variations [26]), and also has some connection with Brenier's weak solutions of incompressible Euler equations [13,41], with Kantorovich's viewpoint in the theory of optimal transportation [75,95] and with Mather's theory [27,90,91]. In order to study the existence, uniqueness and stability (with respect to perturbations of the data) of solutions to the ODE, we consider suitable measures in the space of continuous maps, allowing for the superposition of trajectories.…”

“…The idea of a probabilistic representation is of course classical, and appears in many contexts (particularly for equations of diffusion type). To the best of our knowledge the first reference in the context of conservation laws and fluid mechanics is [41], where a similar approach is proposed for the incompressible Euler equation (see also [42][43][44]): in this case the compact (but neither metrizable, nor separable) space X [0,T ] , with X ⊂ R d compact, was considered.…”

A Family of Stationary Solutions to the Euler Equations and Generalized Solutions

“…of different ways as in [4], [6], [12] and [13]. In [4], was used a concept that takes into account the dynamics of the particles.…”

“…In [4], was used a concept that takes into account the dynamics of the particles. To each path t ∈ [0, T] −→ z(t) ∈ D, one associates the probability that it is followed by some material particle.…”

“…In [4], Brenier shows explicit examples of non trivial generalized solutions. A typical example is when D is the unit disk in 2D and h(x) = −x.…”