Ⅰ Balance Laws 1
1.1 Formulation of the Balance Law 2
1.2 Reduction to Field Equations 3
1.3 Change of Coordinates and a Trace Theorem 7
1.4 Systems of Balance Laws 12
1.5 Companion Balance Laws 13
1.6 Weak and Shock Fronts 15
1.7 Survey of the Theory of BV Functions 17
1.8 BV Solutions of Systems of Balance Laws 21
1.9 Rapid Oscillations and the Stabilizing Effect of Companion Balance Laws 22
1.10 Notes 23
Ⅱ Introduction to Continuum Physics 25
2.1 Bodies and Motions 25
2.2 Balance Laws in Continuum Physics 28
2.3 The Balance Laws of Continuum Thermomechanics 31
2.4 Material Frame Indifference 35
2.5 Thermoelasticity 36
2.6 Thermoviscoelasticity 44
2.7 Incompressibility 47
2.8 Relaxation 48
2.9 Notes 49
Ⅲ Hyperbolic Systems of Balance Laws 53
3.1 Hyperbolicity 53
3.2 Entropy-Entropy Flux Pairs 54
3.3 Examples of Hyperbolic Systems of Balance Laws 56
3.4 Notes 72
Ⅳ The Cauchy Problem 75
4.1 The Cauchy Problem:Ciassical Solutions 75
4.2 Breakdown ofClassical Solutions 78
4.3 The Cauchy Problem:Weak Solutions 81
4.4 Nonuniqueness of Weak Solutions 82
4.5 Entropy Admissibility Condition 83
4.6 The Vanishing Viscosity Approach 87
4.7 Initial-Boundary Value Problems 91
4.8 Notes 95
Ⅴ Entropy and the Stability of Classical Solutions 97
5.1 Convex Entropy and the Existence of Classical Solutions 98
5.2 The Role ofDamping and Relaxation 108
5.3 Convex Entropy and the Stability of Classical Solutions 116
5.4 Involutions 119
5.5 Contingent Entropies and Polyconvexity 129
5.6 Initial-Boundary Value Problems 138
5.7 Notes 141
Ⅵ The L1 Theory for Scalar Conservation Laws 145
6.1 The Cauchy Problem:Perseverance and Demise of Classical Solutions 146
6.2 Admissible Weak Solutions and their Stability Properties 148
6.3 The Method of Vanishing Viscosity 153
6.4 Solutions as Trajectories of a Contraction Semigroup 158
6.5 The Layering Method 164
6.6 Relaxation 167
6.7 A Kinetic Formulation 174
6.8 Fine Structure of L∞ Solutions 180
6.9 Initial-Boundary Value Problems 183
6.10 The L1 Theory for Systems of Conservation Laws 188
6.11 Notes 192
Ⅶ Hyperbolic Systems of Balance Laws in One-Space Dimension 195
7.1 Balance Laws in One-Space Dimension 195
7.2 Hyperbolicity and Strict Hyperbolicity 203
7.3 Riemann Invariants 206
7.4 Entropy-Entropy Flux Pairs 211
7.5 Genuine Nonlinearity and Linear Degeneracy 214
7.6 Simple Waves 216
7.7 Explosion of Weak Fronts 220
7.8 Existence and Breakdown of Classical Solutions 221
7.9 Weak Solutions 225
7.10 Notes 226
Ⅷ Admissible Shocks 231
8.1 Strong Shocks,Weak Shocks,and Shocks of Moderate Strength 231
8.2 The Hugoniot Locus 234
8.3 The Lax Shock Admissibility Criterion;Compressive,Overcompressive and Undercompressive Shocks 240
8.4 The Liu Shock Admissibility Criterion 246
8.5 The Entropy Shock Admissibility Criterion 248
8.6 Viscous Shock Profiles 252
8.7 Nonconservative Shocks 264
8.8 Notes 265
Ⅸ Admissible Wave Fans and the Riemann Problem 271
9.1 Self-Similar Solutions and the Riemann Problem 271
9.2 Wave Fan Admissibility Criteria 274
9.3 Solution of the Riemann Problem via Wave Curves 275
9.4 Systems with Genuinely Nonlinear or Linearly Degenerate Characteristic Families 278
9.5 General Strictly Hyperbolic Systems 283
9.6 Failure of Existence or Uniqueness;Delta Shocks and Transitional Waves 287
9.7 The Entropy Rate Admissibility Criterion 290
9.8 Viscous Wave Fans 299
9.9 Interaction of Wave Fans 309
9.10 Breakdown of Weak Solutions 317
9.11 Notes 320
Ⅹ Generalized Characteristics 325
10.1 BV Solutions 325
10.2 Generalized Characteristics 326
10.3 Extremal Backward Characteristics 328
10.4 Notes 330
Ⅺ Genuinely Nonlinear Scalar Conservation Laws 331
11.1 Admissible BV Solutions and Generalized Characteristics 332
11.2 The Spreading of Rarefaction Waves 335
11.3 Regularity of Solutions 336
11.4 Divides,Invariants and the Lax Formula 340
11.5 Decay of Solutions Induced by Entropy Dissipation 344
11.6 Spreading of Characteristics and Development of N-Waves 346
11.7 Confinement of Characteristics and Formation of Saw-toothed Profiles 348
11.8 Comparison Theorems and L1 Stability 350
11.9 Genuinely Nonlinear Scalar Balance Laws 358
11.10 Balance Laws with Linear Excitation 362
11.11 An Inhomogeneous Conservation Law 365
11.12 Notes 370
Ⅻ Genuinely Nonlinear Systems of Two Conservation Laws 373
12.1 Notation and Assumptions 373
12.2 Entropy-Entropy Flux Pairs and the Hodograph Transformation 375
12.3 Local Structure of Solutions 378
12.4 Propagation of Riemann Invariants Along Extremal Backward Characteristics 381
12.5 Bounds on Solutions 398
12.6 Spreading of Rarefaction Waves 410
12.7 Regularity of Solutions 415
12.8 Initial Data in L1 417
12.9 Initial Data with Compact Support 421
12.10 Periodic Solutions 427
12.11 Notes 432
ⅩⅢ The Random Choice Method 435
13.1 The Construction Scheme 435
13.2 Compactness and Consistency 438
13.3 Wave Interactions,Approximate Conservation Laws and Approximate Characteristics in Genuinely Nonlinear Systems 444
13.4 The Glimm Functional for Genuinely Nonlinear Systems 448
13.5 Bounds on the Total Variation for Genuinely Nonlinear Systems 453
13.6 Bounds on the Supremum for Genuinely Nonlinear Systems 455
13.7 General Systems 457
13.8 Wave Tracing 460
13.9 Inhomogeneous Systems of Balance Laws 463
13.10 Notes 474
ⅩⅣ The Front Tracking Method and Standard Riemann Semigroups 477
14.1 Front Tracking for Scalar Conservation Laws 478
14.2 Front Tracking for Genuinely Nonlinear Systems of Conservation Laws 480
14.3 The Global Wave Pattern 485
14.4 Approximate Solutions 486
14.5 Bounds on the Total Variation 488
14.6 Bounds on the Combined Strength of Pseudoshocks 491
14.7 Compactness and Consistency 494
14.8 Continuous Dependence on Initial Data 496
14.9 The Standard Riemann Semigroup 500
14.10 Uniqueness of Solutions 501
14.11 Continuous Glimm Functionals,Spreading of Rarefaction Waves,and Structure of Solutions 507
14.12 Stability of Strong Waves 510
14.13 Notes 512
ⅩⅤ Construction of BV Solutions by the Vanishing Viscosity Method 517
15.1 The Main Result 517
15.2 Road Map to the ProofofTheorem 15.1.1 519
15.3 The Effects of Diffusion 521
15.4 Decomposition into Viscous Traveling Waves 524
15.5 Transversal Wave Interactions 528
15.6 Interaction of Waves of the Same Family 532
15.7 Energy Estimates 536
15.8 Stability Estimates 539
15.9 Notes 542
ⅩⅥ Compensated Compactness 545
16.1 The Young Measure 546
16.2 Compensated Compactness and the div-curl Lemma 547
16.3 Measure-Valued Solutions for Systems of Conservation Laws and Compensated Compactness 548
16.4 Scalar Conservation Laws 551
16.5 A Relaxation Scheme for Scalar Conservation Laws 553
16.6 Genuinely Nonlinear Systems of Two Conservation Laws 556
16.7 The System of Isentropic Elasticity 559
16.8 The System of Isentropic Gas Dynamics 564
16.9 Notes 567
ⅩⅦ Conservation Laws in Two Space Dimensions 573
17.1 Self-Similar Solutions for Multidimensional Scalar Conservation Laws 573
17.2 Steady Planar Isentropic Gas Flow 576
17.3 Self-Similar Planar Irrotational Isentropic Gas Flow 580
17.4 Supersonic Isentropic Gas Flow Past a Ramp of Gentle Slope 583
17.5 Regular Shock Reflection on a Wall 588
17.6 Shock Collision with a Steep Ramp 591
17.7 Notes 594
Bibliography 597
Author Index 693
Subject Index 703