Embedding Camassa-Holm equations in incompressible Euler

Natale Andrea; Vialard François-Xavier

doi:10.3934/jgm.2019011

Article Contents

2019, Volume 11, Issue 2: 205-223. Doi: 10.3934/jgm.2019011

This issue Previous Article Conservation laws in discrete geometry Next Article Dual pairs and regularization of Kummer shapes in resonances

Embedding Camassa-Holm equations in incompressible Euler

Natale Andrea^1, and
Vialard François-Xavier^2, ,

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
^* Corresponding author: François-Xavier Vialard

Received: March 31, 2018

Revised: February 28, 2019

Published: May 2019

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. A diagram illustrating the key relations leading of the embedding of $ H^{{\rm{div}}} $ geodesic equations into incompressible Euler.

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Figure 2. 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.

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