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Hyperbolic Systems of Balance Laws

Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 14-21, 2003

  • Book
  • © 2007

Overview

Authors:
  1. Alberto Bressan

    1. Department of Mathematics, Penn State University, University Park, USA

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  2. Denis Serre

    1. Unité de mathématiques pures et appliquées, Ecole Normale Supérieure de Lyon, Lyon Cedex 07, France

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  3. Mark Williams

    1. Department of Mathematics, University of North Carolina, Chapel Hill, USA

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  4. Kevin Zumbrun

    1. Department of Mathematics, Indiana University, Bloomington

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Editors:
  1. Pierangelo Marcati

    1. Department of Pure and Applied Mathematics, University of L'Aquila, L'Aquila, Italy

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Part of the book series: Lecture Notes in Mathematics (LNM, volume 1911)

Part of the book sub series: C.I.M.E. Foundation Subseries (LNMCIME)

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Table of contents (4 chapters)

  1. Front Matter

    Pages I-XII
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  2. BV Solutions to Hyperbolic Systems by Vanishing Viscosity

    • Alberto Bressan
    Pages 1-77

  3. Discrete Shock Profiles: Existence and Stability

    • Denis Serre
    Pages 79-158

  4. Stability of Multidimensional Viscous Shocks

    • Mark Williams
    Pages 159-227

  5. Planar Stability Criteria for Viscous Shock Waves of Systems with Real Viscosity

    • Kevin Zumbrun
    Pages 229-326

  6. Back Matter

    Pages 327-356
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Keywords

About this book

The present Cime volume includes four lectures by Bressan, Serre, Zumbrun and Williams and an appendix with a Tutorial on Center Manifold Theorem by Bressan. Bressan’s notes start with an extensive review of the theory of hyperbolic conservation laws. Then he introduces the vanishing viscosity approach and explains clearly the building blocks of the theory in particular the crucial role of the decomposition by travelling waves. Serre focuses on existence and stability for discrete shock profiles, he reviews the existence both in the rational and in the irrational cases and gives a concise introduction to the use of spectral methods for stability analysis. Finally the lectures by Williams and Zumbrun deal with the stability of multidimensional fronts. Williams’ lecture describes the stability of multidimensional viscous shocks: the small viscosity limit, linearization and conjugation, Evans functions, Lopatinski determinants etc. Zumbrun discusses planar stability for viscous shocks with a realistic physical viscosity, necessary and sufficient conditions for nonlinear stability, in analogy to the Lopatinski condition obtained by Majda for the inviscid case.

Authors, Editors and Affiliations

  • Department of Pure and Applied Mathematics, University of L'Aquila, L'Aquila, Italy

    Pierangelo Marcati

  • Department of Mathematics, Penn State University, University Park, USA

    Alberto Bressan

  • Unité de mathématiques pures et appliquées, Ecole Normale Supérieure de Lyon, Lyon Cedex 07, France

    Denis Serre

  • Department of Mathematics, University of North Carolina, Chapel Hill, USA

    Mark Williams

  • Department of Mathematics, Indiana University, Bloomington

    Kevin Zumbrun

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