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Global Semigroup of Conservative Solutions of the Nonlinear Variational Wave Equation

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Abstract

We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation u tt − c(u)(c(u)u x ) x  = 0. We allow for initial data u|t = 0 and u t |t=0 that contain measures. We assume that \({0 < \kappa^{-1} \leqq c(u) \leqq \kappa}\). Solutions of this equation may experience concentration of the energy density \({(u_t^2+c(u)^2u_x^2){\rm d}x}\) into sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby singularities in the energy density become manageable. Furthermore, we prove that the energy may focus only on a set of times of zero measure or at points where c′(u) vanishes. A new numerical method for constructing conservative solutions is provided and illustrated with examples.

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Acknowledgements

We are very indebted to Katrin Grunert for careful reading of the manuscript.

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This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    Helge Holden

  2. Centre of Mathematics for Applications, University of Oslo, 0316, Oslo, Norway

    Helge Holden & Xavier Raynaud

Authors
  1. Helge Holden
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  2. Xavier Raynaud
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Correspondence to Helge Holden.

Additional information

Communicated by Y. Brenier

The research is supported in part by the Research Council of Norway. This paper was written as part of the international research program on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during the academic year 2008–09.

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Holden, H., Raynaud, X. Global Semigroup of Conservative Solutions of the Nonlinear Variational Wave Equation. Arch Rational Mech Anal 201, 871–964 (2011). https://doi.org/10.1007/s00205-011-0403-5

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  • DOI: https://doi.org/10.1007/s00205-011-0403-5

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