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Some Qualitative Properties of 2 × 2 Systems of Conservation Laws of Mixed Type

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Nonlinear Evolution Equations That Change Type

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

Abstract

We study qualitative features of the initial value problemz t + F(z) x = 0, z(x, 0) = z 0 (x), x ∈ R, where z(x,t) ∈ R 2, with Riemann inital data, viz. z 0 (x) = z l if x < 0 and z 0 (x) = z r if x > 0. In particular we are interested in the case when the system changes type when the eigenvalues of the Jacobian dF become complex. It is proved that if z l and z r are in the elliptic region, and the elliptic region is convex, then part of the solution has to be outside the elliptic region. If both z l and z r are in the hyperbolic region, then the solution will not enter the elliptic region. We show with an explicit example that the latter property is not true for general Cauchy data. This example is investigated numerically.

Supported in part by Vista, NAVF and NSF grant DMS-8801918. H.H. would like to thank Barbara Keyfitz, Michael Shearer and the Institute for Mathematics and its Applications for organizing a very stimulating workshop and for the invitation to present these results.

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Authors and Affiliations

  1. Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA, 91125, USA

    H. Holden

  2. Institute of Mathematics, University of Trondheim, N-7034, Trondheim-NTH, Norway

    H. Holden

  3. Norwegian Computing Center, P.O. Box 114, Blindern, N-0314, Oslo 3, Norway

    L. Holden

  4. Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N-0316, Oslo 3, Norway

    N. H. Risebro

Authors
  1. H. Holden
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  2. L. Holden
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  3. N. H. Risebro
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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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Holden, H., Holden, L., Risebro, N.H. (1990). Some Qualitative Properties of 2 × 2 Systems of Conservation Laws of Mixed Type. 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_5

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  • DOI: https://doi.org/10.1007/978-1-4613-9049-7_5

  • 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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