Hyperbolic Systems of Balance Laws

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

  • Alberto Bressan
  • Denis Serre
  • Mark Williams
  • Kevin Zumbrun
  • Pierangelo Marcati

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

Table of contents

  1. Front Matter
    Pages I-XII
  2. Mark Williams
    Pages 159-227
  3. Back Matter
    Pages 327-356

About this book

Introduction

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.

Keywords

Profil discrete shock profiles hyperbolic conservation laws nonlinear hypebolic systems stability of shock waves vanishing viscosity

Authors and affiliations

  • Alberto Bressan
    • 1
  • Denis Serre
    • 2
  • Mark Williams
    • 3
  • Kevin Zumbrun
    • 4
  1. 1.Department of MathematicsPenn State UniversityUniversity ParkUSA
  2. 2.Unité de mathématiques pures et appliquéesEcole Normale Supérieure de LyonLyon Cedex 07France
  3. 3.Department of MathematicsUniversity of North CarolinaChapel HillUSA
  4. 4.Department of MathematicsIndiana UniversityBloomington

Editors and affiliations

  • Pierangelo Marcati
    • 1
  1. 1.Department of Pure and Applied MathematicsUniversity of L'AquilaL'AquilaItaly

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-540-72187-1
  • Copyright Information Springer-Verlag Berlin Heidelberg 2007
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-540-72186-4
  • Online ISBN 978-3-540-72187-1
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • About this book