Generalized compressible fluid flows and solutions of the Camassa-Holm variational model

Abstract

The Camassa-Holm equation on a domain M in Rd, in one of its possible multi-dimensional generalizations, describes geodesics on the group of diffeomorphisms with respect to the H(div) metric. It has been recently reformulated as a geodesic equation for the L2 metric on a subgroup of the diffeomorphism group of the cone over M. We use such an interpretation to construct an analogue of Brenier’s generalized incompressible Euler flows for the Camassa-Holm equation. This involves describing the fluid motion using probability measures on the space of paths on the cone, so that particles are allowed to split and cross. Differently from Brenier’s model, however, we are also able to account for compressibility by employing an explicit probabilistic representation of the Jacobian of the flow map. We formulate the boundary value problem associated to the Camassa-Holm equation using such generalized flows. We prove existence of solutions and that, for short times, smooth solutions of the Camassa-Holm equations are the unique solutions of our model. We propose a numerical scheme to construct generalized solutions on the cone and present some numerical results illustrating the relation between the generalized Camassa-Holm and incompressible Euler solutions.