Hyperbolic Conservation Laws in Continuum Physics

Preface to the Fourth Edition......Page 7
Acknowledgments......Page 9
Introduction......Page 10
A Sketch of the Early History of Hyperbolic Conservation Laws......Page 15
Contents......Page 31
IBalance Laws......Page 37
 1.1 Formulation of the Balance Law......Page 38
 1.2 Reduction to Field Equations......Page 39
 1.3 Change of Coordinates and a Trace Theorem......Page 43
 1.4 Systems of Balance Laws......Page 48
 1.5 Companion Balance Laws......Page 49
 1.6 Weak and Shock Fronts......Page 51
 1.7 Survey of the Theory of BV Functions......Page 53
 1.8 BV Solutions of Systems of Balance Laws......Page 57
 1.10 Notes......Page 59
 2.1 Kinematics......Page 61
 2.2 Balance Laws in Continuum Physics......Page 64
 2.3 The Balance Laws of Continuum Thermomechanics......Page 67
 2.4 Material Frame Indifference......Page 71
 2.5 Thermoelasticity......Page 72
 2.6 Thermoviscoelasticity......Page 80
 2.7 Incompressibility......Page 83
 2.8 Relaxation......Page 84
 2.9 Notes......Page 85
 3.1 Hyperbolicity......Page 88
 3.2 Entropy-Entropy Flux Pairs......Page 89
 3.3 Examples of Hyperbolic Systems of Balance Laws......Page 91
 3.4 Notes......Page 108
 4.1 The Cauchy Problem: Classical Solutions......Page 111
 4.2 Breakdown of Classical Solutions......Page 114
 4.3 The Cauchy Problem:Weak Solutions......Page 116
 4.4 Nonuniqueness ofWeak Solutions......Page 117
 4.5 Entropy Admissibility Condition......Page 118
 4.6 The Vanishing Viscosity Approach......Page 124
 4.7 Initial-Boundary Value Problems......Page 128
 4.8 Euler Equations......Page 131
 4.9 Notes......Page 141
VEntropy and the Stability of Classical Solutions......Page 144
 5.1 Convex Entropy and the Existence of Classical Solutions......Page 145
 5.2 Relative Entropy and the Stability of Classical Solutions......Page 155
 5.3 Involutions and Contingent Entropies......Page 158
 5.4 Contingent Entropies and Polyconvexity......Page 171
 5.5 The Role of Damping and Relaxation......Page 179
 5.6 Initial-Boundary Value Problems......Page 193
 5.7 Notes......Page 203
VIThe L1 Theory for Scalar Conservation Laws......Page 208
 6.1 The Cauchy Problem: Perseverance and Demise of Classical Solutions......Page 209
 6.2 AdmissibleWeak Solutions and their Stability Properties......Page 211
 6.3 The Method of Vanishing Viscosity......Page 216
 6.4 Solutions as Trajectories of a Contraction Semigroup and the Large Time Behavior of Periodic Solutions......Page 221
 6.5 The Layering Method......Page 228
 6.6 Relaxation......Page 232
 6.7 A Kinetic Formulation......Page 238
 6.8 Fine Structure of L∞ Solutions......Page 245
 6.9 Initial-Boundary Value Problems......Page 248
 6.10 The L1 Theory for Systems of Conservation Laws......Page 253
 6.11 Notes......Page 256
 7.1 Balance Laws in One-Space Dimension......Page 260
 7.2 Hyperbolicity and Strict Hyperbolicity......Page 268
 7.3 Riemann Invariants......Page 271
 7.4 Entropy-Entropy Flux Pairs......Page 276
 7.5 Genuine Nonlinearity and Linear Degeneracy......Page 278
 7.6 SimpleWaves......Page 280
 7.7 Explosion ofWeak Fronts......Page 285
 7.8 Existence and Breakdown of Classical Solutions......Page 286
 7.9 Weak Solutions......Page 290
 7.10 Notes......Page 291
 8.1 Strong Shocks,Weak Shocks, and Shocks of Moderate Strength......Page 295
 8.2 The Hugoniot Locus......Page 298
 8.3 The Lax Shock Admissibility Criterion; Compressive, Overcompressive and Undercompressive Shocks......Page 304
 8.4 The Liu Shock Admissibility Criterion......Page 310
 8.5 The Entropy Shock Admissibility Criterion......Page 312
 8.6 Viscous Shock Profiles......Page 317
 8.7 Nonconservative Shocks......Page 328
 8.8 Notes......Page 329
 9.1 Self-Similar Solutions and the Riemann Problem......Page 335
 9.2 Wave Fan Admissibility Criteria......Page 339
 9.3 Solution of the Riemann Problem viaWave Curves......Page 341
 9.4 Systems with Genuinely Nonlinear or Linearly Degenerate Characteristic Families......Page 344
 9.5 General Strictly Hyperbolic Systems......Page 348
 9.6 Failure of Existence or Uniqueness; Delta Shocks and TransitionalWaves......Page 352
 9.7 The Entropy Rate Admissibility Criterion......Page 355
 9.8 ViscousWave Fans......Page 364
 9.9 Interaction ofWave Fans......Page 375
 9.10 Breakdown ofWeak Solutions......Page 382
 9.11 Notes......Page 386
 10.1 BV Solutions......Page 391
 10.2 Generalized Characteristics......Page 392
 10.3 Extremal Backward Characteristics......Page 394
 10.4 Notes......Page 397
XIScalar Conservation Laws in One Space Dimension......Page 398
 11.1 Admissible BV Solutions and Generalized Characteristics......Page 399
 11.2 The Spreading of RarefactionWaves......Page 402
 11.3 Regularity of Solutions......Page 403
 11.4 Divides, Invariants and the Lax Formula......Page 408
 11.5 Decay of Solutions Induced by Entropy Dissipation......Page 411
 11.6 Spreading of Characteristics and Development of N-Waves......Page 414
 11.7 Confinement of Characteristics and Formation of Saw-toothed Profiles......Page 415
 11.8 Comparison Theorems and L1 Stability......Page 417
 11.9 Genuinely Nonlinear Scalar Balance Laws......Page 426
 11.10 Balance Laws with Linear Excitation......Page 430
 11.11 An Inhomogeneous Conservation Law......Page 432
 11.12 When Genuine Nonlinearity Fails......Page 437
 11.13 Entropy Production......Page 449
 11.14 Notes......Page 453
 12.1 Notation and Assumptions......Page 458
 12.2 Entropy-Entropy Flux Pairs and the Hodograph Transformation......Page 460
 12.3 Local Structure of Solutions......Page 463
 12.4 Propagation of Riemann Invariants Along Extremal Backward Characteristics......Page 466
 12.5 Bounds on Solutions......Page 483
 12.6 Spreading of RarefactionWaves......Page 495
 12.7 Regularity of Solutions......Page 500
 12.8 Initial Data in L1......Page 502
 12.9 Initial Data with Compact Support......Page 506
 12.10 Periodic Solutions......Page 512
 12.11 Notes......Page 517
 13.1 The Construction Scheme......Page 519
 13.2 Compactness and Consistency......Page 522
 13.3 Wave Interactions in Genuinely Nonlinear Systems......Page 528
 13.4 The Glimm Functional for Genuinely Nonlinear Systems......Page 530
 13.5 Bounds on the Total Variation for Genuinely Nonlinear Systems......Page 535
 13.6 Bounds on the Supremum for Genuinely Nonlinear Systems......Page 537
 13.7 General Systems......Page 539
 13.8 Wave Tracing......Page 542
 13.9 Notes......Page 545
XIVThe Front Tracking Method and Standard Riemann Semigroups......Page 547
 14.1 Front Tracking for Scalar Conservation Laws......Page 548
 14.2 Front Tracking for Genuinely Nonlinear Systems of Conservation Laws......Page 550
 14.3 The GlobalWave Pattern......Page 555
 14.4 Approximate Solutions......Page 556
 14.5 Bounds on the Total Variation......Page 558
 14.6 Bounds on the Combined Strength of Pseudoshocks......Page 561
 14.7 Compactness and Consistency......Page 564
 14.8 Continuous Dependence on Initial Data......Page 566
 14.9 The Standard Riemann Semigroup......Page 570
 14.10 Uniqueness of Solutions......Page 571
 14.11 Continuous Glimm Functionals, Spreading of RarefactionWaves, and Structure of Solutions......Page 577
 14.12 Stability of StrongWaves......Page 580
 14.13 Notes......Page 582
 15.1 The Main Result......Page 586
 15.2 Road Map to the Proof of Theorem 15.1.1......Page 588
 15.3 The Effects of Diffusion......Page 590
 15.4 Decomposition into Viscous TravelingWaves......Page 593
 15.5 TransversalWave Interactions......Page 597
 15.6 Interaction ofWaves of the Same Family......Page 601
 15.7 Energy Estimates......Page 605
 15.8 Stability Estimates......Page 608
 15.9 Notes......Page 611
XVIBV Solutions for Systems of Balance Laws......Page 613
 16.1 The Cauchy Problem......Page 614
 16.2 Strong Dissipation......Page 617
 16.3 Redistribution of Damping......Page 621
 16.4 Bounds on the Variation......Page 623
 16.5 L1 Stability Via Entropy with Conical Singularity at the Origin......Page 634
 16.6 L1 Stability when the Source is Partially Dissipative......Page 637
 16.7 Notes......Page 650
XVIICompensated Compactness......Page 651
 17.1 The Young Measure......Page 652
 17.2 Compensated Compactness and the div-curl Lemma......Page 653
 17.3 Measure-Valued Solutions for Systems of Conservation Laws and Compensated Compactness......Page 654
 17.4 Scalar Conservation Laws......Page 657
 17.5 A Relaxation Scheme for Scalar Conservation Laws......Page 659
 17.6 Genuinely Nonlinear Systems of Two Conservation Laws......Page 662
 17.7 The System of Isentropic Elasticity......Page 665
 17.8 The System of Isentropic Gas Dynamics......Page 670
 17.9 Notes......Page 676
 18.1 Self-Similar Solutions for Multidimensional Scalar Conservation Laws......Page 682
 18.2 Steady Planar Isentropic Gas Flow......Page 685
 18.3 Self-Similar Planar Irrotational Isentropic Gas Flow......Page 690
 18.4 Supersonic Isentropic Gas Flow Past a Ramp......Page 694
 18.5 Regular Shock Reflection on aWall......Page 699
 18.6 Shock Collision with a Ramp......Page 702
 18.7 Isometric Immersions......Page 705
 18.8 Cavitation in Elastodynamics......Page 709
 18.9 Notes......Page 713
Bibliography......Page 717
Author Index......Page 837
Subject Index......Page 847