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An Existence and Uniqueness Result for Two Nonstrictly Hyperbolic Systems

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Book cover Nonlinear Evolution Equations That Change Type

Part of the book series: The IMA Volumes in Mathematics and Its Applications ((IMA,volume 27))

Abstract

We prove a result of existence and uniqueness of entropy weak solutions for two nonstrictly hyperbolic systems, both a nonconservative system of two equations

$$ {\partial_t}u + {\partial_x}f(u) = 0,\,{\partial_t}w + a(u){\partial_x}w = 0 $$

, and a conservative system of two equations

$$ {\partial_t}u + {\partial_x}f(u) = 0,\,{\partial_t}v + {\partial_x}(a(u)v) = 0 $$

, where f: R → R is a given strictly convex function and \( a = \frac{d}{{du}}f \). We use the Volpert’s product ([19], see also Dal Maso — Le Floch — Murat [1]) and find entropy weak solutions u and w which have bounded variation while the solutions v are Borel measures. The equations for w and v can be viewed as linear hyperbolic equations with discontinuous coefficients.

This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation.

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References

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Author information

Authors and Affiliations

  1. Centre de Mathématiques Appliquées, URAD0756, Ecole Polytechnique, 91128, Palaiseau Cedex, France

    Philippe Le Floch

Authors
  1. Philippe Le Floch
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, Texas, 77204, USA

    Barbara Lee Keyfitz

  2. Department of Mathematics, North Carolina State University, Raleigh, North Carolina, 27695, USA

    Michael Shearer

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© 1990 Springer-Verlag New York Inc.

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Le Floch, P. (1990). An Existence and Uniqueness Result for Two Nonstrictly Hyperbolic Systems. In: Keyfitz, B.L., Shearer, M. (eds) Nonlinear Evolution Equations That Change Type. The IMA Volumes in Mathematics and Its Applications, vol 27. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9049-7_10

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  • DOI: https://doi.org/10.1007/978-1-4613-9049-7_10

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-9051-0

  • Online ISBN: 978-1-4613-9049-7

  • eBook Packages: Springer Book Archive

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