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Generalized Compressible Flows and Solutions of the \(H(\mathrm {div})\) Geodesic Problem

  • Thomas O. Gallouët
  • Andrea NataleEmail author
  • François-Xavier Vialard
Article
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Abstract

We study the geodesic problem on the group of diffeomorphism of a domain \(M\subset {\mathbb {R}}^d\), equipped with the \(H(\mathrm {div})\) 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 \(H(\mathrm {div})\) 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.

Notes

Acknowledgements

The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA Grant agreement n. PCOFUND-GA-2013-609102, through the PRESTIGE programme coordinated by Campus France. The authors would also like to acknowledge the support from the Project MAGA ANR-16-CE40-0014 (2016-2020).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Thomas O. Gallouët
    • 1
  • Andrea Natale
    • 1
    Email author
  • François-Xavier Vialard
    • 2
  1. 1.InriaParisFrance
  2. 2.Université Paris-Est Marne-la-ValléeLIGMMarne-la-ValléeFrance

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