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Birkhäuser

Advances in the Theory of Shock Waves

  • Book
  • © 2001

Overview

Authors:
  1. Tai-Ping Liu
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  2. Guy Métivier
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  3. Joel Smoller
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  4. Blake Temple
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  5. Wen-An Yong
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  6. Kevin Zumbrun
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Editors:
  1. Heinrich Freistühler

    1. Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

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  2. Anders Szepessy

    1. Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden

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Part of the book series: Progress in Nonlinear Differential Equations and Their Applications (PNLDE, volume 47)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-viii
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  2. Well-Posedness Theory for Hyperbolic Systems of Conservation Laws

    • Tai-Ping Liu
    Pages 1-24

  3. Stability of Multidimensional Shocks

    • Guy Métivier
    Pages 25-103

  4. Shock Wave Solutions of the Einstein Equations: A General Theory with Examples

    • Joel Smoller, Blake Temple
    Pages 105-258

  5. Basic Aspects of Hyperbolic Relaxation Systems

    • Wen-An Yong
    Pages 259-305

  6. Multidimensional Stability of Planar Viscous Shock Waves

    • Kevin Zumbrun
    Pages 307-516

  7. Back Matter

    Pages 517-520
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Keywords

About this book

In the field known as "the mathematical theory of shock waves," very exciting and unexpected developments have occurred in the last few years. Joel Smoller and Blake Temple have established classes of shock wave solutions to the Einstein­ Euler equations of general relativity; indeed, the mathematical and physical con­ sequences of these examples constitute a whole new area of research. The stability theory of "viscous" shock waves has received a new, geometric perspective due to the work of Kevin Zumbrun and collaborators, which offers a spectral approach to systems. Due to the intersection of point and essential spectrum, such an ap­ proach had for a long time seemed out of reach. The stability problem for "in­ viscid" shock waves has been given a novel, clear and concise treatment by Guy Metivier and coworkers through the use of paradifferential calculus. The L 1 semi­ group theory for systems of conservation laws, itself still a recent development, has been considerably condensed by the introduction of new distance functionals through Tai-Ping Liu and collaborators; these functionals compare solutions to different data by direct reference to their wave structure. The fundamental prop­ erties of systems with relaxation have found a systematic description through the papers of Wen-An Yong; for shock waves, this means a first general theorem on the existence of corresponding profiles. The five articles of this book reflect the above developments.

Editors and Affiliations

  • Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

    Heinrich Freistühler

  • Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden

    Anders Szepessy

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