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内容简介
TheseedsofContinuumPhysicswereplantedwiththeworksofthenaturalphilosophersoftheeighteenthcentury,mostnotablyEuler,bythemid-nineteenthcentury,thetreeswerefullygrownandreadytoyieldfruit.Itwasinthisenvironmentthatthestudyofgasdynamicsgavebirthtothetheoryofquasilinearhyperbolicsystemsindivergenceform,commonlycalled“hyperbolicconservationlaws”;andthesetwosubjecthavebeentravelinghand-in-handoverthepastonehundredandfiftyyears.ThisbookaimsatpresentingthetheoryofhyperbolicconservationlawsfromthestandpointofitsgeneticrelationtoContinuumPhysics.Eventhoughresearchisstillmarchingatabriskpace,bothfieldshaveattainedbynowthedegreeofmaturitythatwouldwarrantthewritingofsuchanexposition.
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目录
Chapter I. Balance Laws
1.1 Formulation of the Balance Law
1.2 Reduction to Field Equations
1.3 Change of Coordinates
1.4 Systems of Balance Laws
1.5 Companion Systems of Balance Laws
1.6 Weak and Shock Fronts
1.7 Survey of the Theory of B V Functions
1.8 B V Solutions of Systems of Balance Laws
1.9 Rapid Oscillations and the Stabilizing Effect of Companion Balance Laws
1.1 Notes
Chapter II. Introduction to Continuum Physics
2.1 Bodies and Motions
2.2 Balance Laws in Continuum Physics
2.3 The Balance Laws of Continuum Thermomechanics
2.4 Material Frame Indifference
2.5 Thermoelasticity
2.6 Thermoviscoelasticity
2.7 Notes
Chapter III. Hyperbolic Systems of Balance Laws
3.1 Hyperbolicity
3.2 Entropy-Entropy Flux Pairs
3.3 Examples of Hyperbolic Systems of Balance Laws
3.4 Notes
Chapter IV. The Initial-Value Problem: Admissibility of Solutions ,
4.1 The Initial-Value Problem
4.2 The Burgers Equation and Nonuniqueness of Weak Solutions
4.3 Entropies and Admissible Solutions
4.4 The Vanishing Viscosity Approach
4.5 Initial-Boundary-Value Problems
4.6 Notes
Chapter V. Entropy and the Stability of Classical Solutions
5.1 Convex Entropy and the Existence of Classical Solutions
5.2 Convex Entropy and the Stability of Classical Solutions
5.3 Partially Convex Entropies and Involutions
5.4 Notes
Chapter VI. The L1 Theory of the Scalar Conservation Law
6.1 The Initial-Value Problem: Perseverance and Demise of Classical Solutions
6.2 Admissible Weak Solutions and Their Stability Properties
6.3 The Method of Vanishing Viscosity
6.4 Solutions as Trajectories of a Contraction Semigroup
6.5 The Layering Method
6.6 A Kinetic Formulation
6.7 Relaxation
6.8 The L1 Theory for Systems of Balance Laws
6.9 Notes
Chapter VII. Hyperbolic Systems of Balance Laws in One-Space Dimension
7.1 Balance Laws in One-Space Dimension
7.2 Hyperbolicity and Strict Hyperbolicity
7.3 Riemann Invariants
7.4 Entropy-Entropy Flux Pairs
7.5 Genuine Nonlinearity and Linear Degeneracy
7.6 Simple Waves
7.7 Breakdown of Classical Solutions
7.8 Weak Solutions
7.9 Notes
Chapter VIII. Admissible Shocks
8.1 Strong Shocks, Weak Shocks, and Shocks of Moderate Strength
8.2 The Hugoniot Locus
8.3 The Lax Shock Admissibility Criterion
8.4 The Liu Shock Admissibility Criterion
8.5 The Entropy Shock Admissibility Criterion
8.6 Viscous Shock Profiles
8.7 Notes
Chapter IX. Admissible Wave Fans and the Riemann Problem
9.1 Self-similar Solutions and the Riemann Problem
9.2 Wave Fan Admissibility Criteria
9.3 Solution of the Riemann Problem with Admissible Shocks
9.4 The Entropy Rate Admissibility Criterion
9.5 Viscous Wave Fans
9.6 Interaction of Wave Fans
9.7 Notes
Chapter X. Generalized Characteristics
1.1 B V Solutions
1.2 Generalized Characteristics
1.3 Extremal Backward Characteristics
1.4 Notes
Chapter XI. Genuinely Nonlinear Scalar Conservation Laws
11.1 Admissible B V Solutions and Generalized Characteristics
11.2 The Spreading of Rarefaction Waves
11.3 Regularity of Solutions
11.4 Divides, Invariants and the Lax Formula
11.5 Decay of Solutions Induced by Entropy Dissipation
11.6 Spreading of Characteristics and Development of N-Waves
11.7 Confinement of Characteristics and Formation of Sawtoothed Profiles
11.8 Comparison Theorems and L~ Stability
11.9 Notes
Chapter XII. Genuinely Nonlinear Systems of Two Conservation Law
12.1 Notation and Assumptions
12.2 Entropy-Entropy Flux Pairs
12.3 Local Structure of Solutions
12.4 Propagation of Riemann Invariants Along Extremal Backward Characteristics
12.5 Bounds on Solutions
12.6 Spreading of Rarefaction Waves
12.7 Regularity of Solutions
12.8 Initial Data in Ll
12.9 Initial Data with Compact Support
12.1 Periodic Solutions.
12.11 Notes
Chapter XIII. The Random Choice Method
13.1 The Construction Scheme
13.2 Compactness and Consistency
13.3 Wave Interactions, Approximate Conservation Laws
13.4 The Glimm Functional
13.5 Bounds on the Total Variation
13.6 Bounds on the Supremum
13.7 Wave Partitioning
13.8 Inhomogeneous Systems of Balance Laws
13.9 Breakdown of Weak Solutions
13.1 Notes
Chapter XIV. The Front Tracking Methodand Standard Riemann Semigroups
14.1 The Scalar Conservation Law
14.2 Front Tracking for Systems of Conservation Laws
14.3 The Global Wave Pattern
14.4 Approximate Solutions
14.5 Bounds on the Total Variation
14.6 Bounds on the Combined Strength of Pseudoshocks
14.7 Compactness and Consistency
14.8 Continuous Dependence on Initial Data
14.9 The Standard Riemann Semigroup
14.1 Uniqueness of Solutions
14.11 Structure of Solutions
14.12 Notes
Chapter XV. Compensated Compactness
15.1 The Young Measure.
15.2 Compensated Compactness and the div-curl Lemma
15.3 Measure-Valued Solutions for Systems of Conservation Laws and Compensated Compactness
15.4 Scalar Conservation Laws
15.5 A Relaxation Scheme for Scalar Conservation Laws
15.6 Genuinely Nonlinear Systems of Two Conservation Laws
15.7 The System of Isentropic Elasticity
15.8 The System of Isentropic Gas Dynamics.
15.9 Notes
Bibliography
Author Index
Subject Index
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