# American Institute of Mathematical Sciences

June  2019, 11(2): 205-223. doi: 10.3934/jgm.2019011

## Embedding Camassa-Holm equations in incompressible Euler

 1 INRIA, Paris, 2 Rue Simone Iff, 75012 Paris, France 2 Université Paris-Est Marne-la-Vallée, 5 Boulevard Descartes, 77420 Champs-sur-Marne, France

* Corresponding author: François-Xavier Vialard

Received  April 2018 Revised  March 2019 Published  May 2019

Fund Project: The first author was supported by the People Programme (Marie Curie Actions) of the European Unions Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n. PCOFUND-GA-2013-609102, through the PRESTIGE programme coordinated by Campus France

Recently, Gallouët 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 $b$-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.

Citation: Andrea Natale, François-Xavier Vialard. Embedding Camassa-Holm equations in incompressible Euler. Journal of Geometric Mechanics, 2019, 11 (2) : 205-223. doi: 10.3934/jgm.2019011
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A diagram illustrating the key relations leading of the embedding of $H^{{\rm{div}}}$ geodesic equations into incompressible Euler.
A peakons-antipeakons collision represented in the new polar coordinates variables at different time points. The two curves in blue and green are scaled versions of each others: they represent the image under the map $\Psi(\theta,r) = r \sqrt{\partial_\theta \varphi(\theta)} e^{i\varphi(\theta)}$ of two circles on the complex plane with different radii.