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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