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The Burgers–Hilbert Equation

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Theory, Numerics and Applications of Hyperbolic Problems II (HYP 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 237))

Abstract

The Burgers–Hilbert equation consists of an inviscid Burgers equation with a linear Hilbert-transform source term. We explain how the equation arises as a model for waves on a vorticity discontinuity and surface waves with constant frequency. We survey various results about the Burger–Hilbert equation, including ones on singularity formation, shock structure, weak solutions, and the enhanced life span of small, smooth solutions.

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Notes

  1. 1.

    See [1] for a similar example involving an application of functional methods to the inviscid Burger’s equation.

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Acknowledgements

Supported by the NSF under grant number 1616988.

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Authors and Affiliations

  1. University of California at Davis, Davis, CA, 95616, USA

    John K. Hunter

Authors
  1. John K. Hunter
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Correspondence to John K. Hunter .

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Editors and Affiliations

  1. Department of Mathematics, Würzburg University, Würzburg, Germany

    Christian Klingenberg

  2. Department of Mathematics, RWTH Aachen University, Aachen, Germany

    Michael Westdickenberg

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Hunter, J.K. (2018). The Burgers–Hilbert Equation. In: Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems II. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol 237. Springer, Cham. https://doi.org/10.1007/978-3-319-91548-7_3

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