Systems of partial differential equations reflecting conservation laws hold significant relevance to a variety of theoretical and practical applications, including compressible fluid flow, electromagnetism, elasticity theory, and other areas of continuum mechanics. This field of nonlinear analysis is currently experiencing a marked increase in successful research activity.
The EU-TMR network "Hyperbolic Systems of Conservation Laws held a summer program offering short courses on the Analysis of Systems of Conservation Laws. This book contains five of the self-contained short courses presented during this program by experts of international reputation. These courses, which address solutions to hyperbolic systems by the front tracking method, non-strictly hyperbolic conservation laws, hyperbolic-elliptic coupled systems, hyperbolic relaxation problems, the stability of nonlinear waves in viscous media and numerics, and more, represent the state of the art of most central aspects of the field.
Preface Analysis of Solutions to Hyperbolic Systems by the Front Tracking Method, A. Bressan, Scuola Internazionale Superiore di Studi Avanzati, Trieste A Wave-Front Tracking Algorithm Continuous Dependence on Initial Data A Front Tracking Algorithm for 2 x 2 Systems Uniqueness of the Standard Riemann Semigroup Characterization of Semigroup Trajectories Unique Solutions to the Cauchy Problem References Hyperbolic Conservation Laws, P.T. Kan, Purdue University, Indianapolis, Indiana Global Solutions to Systems with Umbilic Degeneracy Introduction General Classes and a Canonical Family of Degenerate Systems Parabolic Approximation and Young Measures Entropy Functions and Entropy Dissipation Measures Cm Goursat Entropies Convergence of Approximate Solutions Initial Boundary Value Problems in L8 Introduction Boundary Sets Godunov Schemes for IBVP Traces of Entropy Fluxes Godunov Schemes for Scalar Equations: Interior and Boundary Regularity The Study of Boundary Layers A Class of Quadratic Systems and a Class of Well-Posed IBVP References The Initial Value Problem for Hyperbolic-Elliptic Coupled Systems and Applications to Radiation Hydrodynamics, S. Kawashima, Y. Nikkuni, S. Nishibata, Kyushu University, Fukuoka Introduction Entropy Function and Symmetrization Local Existence Stability Condition Global Existence Proof of a Priori Estimate Decay Properties for the Linearized System Decay Estimate Large-Time Approximation Applications References Recent Results on Hyperbolic Relaxation Problems, R. Natalini, Istituto per le Applicazioni del Calcolo, Rome, Italy Introduction Motivations The Smooth Case Discontinuous Equilibrium Solutions and Weak Convergence Methods The BV Framework References Lectures on Stability of Nonlinear Waves in Viscous Media and Numerics, A. Szepessy, Kungl. Tekniska Hoegskolan, Stockholm Introduction Perturbations of Constant States Shock Waves Rarefaction Waves Multigrid Methods for Flow Problems References