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On the Equivalence of Eulerian and Lagrangian Variables for the Two-Component Camassa–Holm System

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Book cover Current Research in Nonlinear Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 135))

Abstract

The Camassa–Holm equation and its two-component Camassa–Holm system generalization both experience wave breaking in finite time. To analyze this, and to obtain solutions past wave breaking, it is common to reformulate the original equation given in Eulerian coordinates, into a system of ordinary differential equations in Lagrangian coordinates. It is of considerable interest to study the stability of solutions and how this is manifested in Eulerian and Lagrangian variables. We identify criteria of convergence, such that convergence in Eulerian coordinates is equivalent to convergence in Lagrangian coordinates. In addition, we show how one can approximate global conservative solutions of the scalar Camassa–Holm equation by smooth solutions of the two-component Camassa–Holm system that do not experience wave breaking.

Dedicated to Haim Brezis and Louis Nirenberg in deep admiration.

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Notes

  1. 1.

    This construction resembles the one used in Step 2 of the proof of Theorem 1. However, here we perform the construction in Lagrangian variables.

  2. 2.

    Note the factors g n,ξ(ξ).

  3. 3.

    Here we denote by y # (h dξ) the push-forward of the measure h dξ by y, defined by \(y_\#(h\,d\xi )(A) = \int _{y^{-1}(A)} h(\xi )d\xi \) for all Borel sets \(A \subset \mathbb {R}\).

  4. 4.

    We say that \(f_n\rightharpoonup f\) if \(\int _{\mathbb {R}} f_n(x)g(x)dx\to \int _{\mathbb {R}} f(x)g(x)dx \) for every \(g\in L^2(\mathbb {R})\).

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Acknowledgements

Research supported in part by the Research Council of Norway projects NoPiMa and WaNP, and by the Austrian Science Fund (FWF) under Grant No. J3147. KG and HH are grateful to Institut Mittag-Leffler, Stockholm, for the generous hospitality during the fall of 2016, when part of this paper was written.

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Authors and Affiliations

  1. Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, Trondheim, Norway

    Markus Grasmair, Katrin Grunert & Helge Holden

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  1. Markus Grasmair
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  2. Katrin Grunert
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  3. Helge Holden
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Correspondence to Helge Holden .

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Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Grasmair, M., Grunert, K., Holden, H. (2018). On the Equivalence of Eulerian and Lagrangian Variables for the Two-Component Camassa–Holm System. In: Rassias, T. (eds) Current Research in Nonlinear Analysis. Springer Optimization and Its Applications, vol 135. Springer, Cham. https://doi.org/10.1007/978-3-319-89800-1_7

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