Embedding Camassa-Holm equations in incompressible Euler

Abstract

Recently, Gallouet and Vialard [ 11 ] showed that the CH equation can be embedded in the incompressible Euler equation on a non compact Riemannian manifold. After surveying this result from a geometric point of view, we extend it to a broader class of PDEs, namely the so-called CH2 equations and the Holm-Staley \begin{document}$b$\end{document} -family of equations. A salient feature of these embeddings is the cone singularity of the Riemannian manifold on which the incompressible Euler equation is considered.