Generalized Compressible Flows and Solutions of the H(div) Geodesic Problem

We study the geodesic problem on the group of diffeomorphism of a domain M⊂ℝ^d , equipped with the H(div) metric. The geodesic equations coincide with the Camassa–Holm equation when d=1 , and represent one of its possible multi-dimensional generalizations when d>1 . We propose a relaxation à la Brenier of this problem, in which solutions are represented as probability measures on the space of continuous paths on the cone over M . We use this relaxation to prove that smooth H(div) geodesics are globally length minimizing for short times. We also prove that there exists a unique pressure field associated to solutions of our relaxation. Finally, 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.