Analysis of Systems of Conservation Laws
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Summary
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.
Table of Contents
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 Högskolan, Stockholm
Introduction
Perturbations of Constant States
Shock Waves
Rarefaction Waves
Multigrid Methods for Flow Problems
References
