First we recall the link between Incompressible Euler Equation and Optimal transport (throughout a riemannian submersion) and some of the results that can be deduced using this geometrical link, for instance Brenier generalized geodesics, Polar projection, Lagrangian numerical scheme. We then aim to explain why the couple [Camassa-Holm equation/$H^{\text{div}}$/Unbalanced Optimal transport] share the same geometrical structure and discuss which of the previous results can be extended in this case. In particular a definition of a pression for the Camassa-Holm equation naturally arises in this framework.
Paul Pegon