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 (auth.), Pierangelo Marcati (eds.)9783540721864, 3-540-72186-X

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.

Table of contents :
Front Matter….Pages I-XII
BV Solutions to Hyperbolic Systems by Vanishing Viscosity….Pages 1-77
Discrete Shock Profiles: Existence and Stability….Pages 79-158
Stability of Multidimensional Viscous Shocks….Pages 159-227
Planar Stability Criteria for Viscous Shock Waves of Systems with Real Viscosity….Pages 229-326
Back Matter….Pages 327-356