Skip to Main content Skip to Navigation
Journal articles

Generalized compressible flows and solutions of the H(div) geodesic problem

Abstract : We study the geodesic problem on the group of diffeomorphism of a domain M⊂Rd, 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.
Complete list of metadata
Contributor : Andrea Natale Connect in order to contact the contributor
Submitted on : Monday, October 7, 2019 - 7:18:15 PM
Last modification on : Friday, February 4, 2022 - 3:12:59 AM


Files produced by the author(s)



Thomas Gallouët, Andrea Natale, François-Xavier Vialard. Generalized compressible flows and solutions of the H(div) geodesic problem. Archive for Rational Mechanics and Analysis, Springer Verlag, In press, ⟨10.1007/s00205-019-01453-x⟩. ⟨hal-01815531v3⟩



Record views


Files downloads