Hyperbolic Conservation Laws in Continuum Physics (eBook)

Professor Dafermos received a Diploma in Civil Engineering from the National Technical University of Athens (1964) and a Ph.D. in Mechanics from the Johns Hopkins University (1967). He has served as Assistant Professor at Cornell University (1968-1971),and as Associate Professor (1971-1975) and Professor (1975- present) in the Division of Applied Mathematics at Brown University. In addition, Professor Dafermos has served as Director of the Lefschetz Center of Dynamical Systems (1988-1993, 2006-2007), as Chairman of the Society for Natural Philosophy (1977-1978) and as Secretary of the International Society for the Interaction of Mathematics and Mechanics. Since 1984, he has been the Alumni-Alumnae University Professor at Brown.In addition to several honorary degrees, he has received the SIAM W.T. and Idalia Reid Prize (2000), the Cataldo e Angiola Agostinelli Prize of the Accademia Nazionale dei Lincei (2011), the Galileo Medal of the City of Padua (2012), and the Prize of the International Society for the Interaction of Mechanics and Mathematics (2014). He was elected a Fellow of SIAM (2009) and a Fellow of the AMS (2013). In 2016 he received the Wiener Prize, awarded jointly by the American Mathematical Society (AMS) and the Society for Industrial and Applied Mathematics (SIAM).

Preface to the Third Edition 6
Introduction 8
A Sketch of the Early History of Hyperbolic Conservation Laws 12
Contents 28
I Balance Laws 33
1.1 Formulation of the Balance Law 34
1.2 Reduction to Field Equations 35
1.3 Change of Coordinates and a Trace Theorem 39
1.4 Systems of Balance Laws 44
1.5 Companion Balance Laws 45
1.6 Weak and Shock Fronts 47
1.7 Survey of the Theory of BV Functions 49
1.8 BV Solutions of Systems of Balance Laws 53
1.9 Rapid Oscillations and the Stabilizing Effect of Companion Balance Laws 54
1.10 Notes 55
II Introduction to Continuum Physics 57
2.1 Bodies and Motions 57
2.2 Balance Laws in Continuum Physics 60
2.3 The Balance Laws of Continuum Thermomechanics 63
2.4 Material Frame Indifference 67
2.5 Thermoelasticity 68
2.6 Thermoviscoelasticity 76
2.7 Incompressibility 79
2.8 Relaxation 80
2.9 Notes 81
III Hyperbolic Systems of Balance Laws 84
3.1 Hyperbolicity 84
3.2 Entropy-Entropy Flux Pairs 85
3.3 Examples of Hyperbolic Systems of Balance Laws 87
3.4 Notes 103
IV The Cauchy Problem 106
4.1 The Cauchy Problem: Classical Solutions 106
4.2 Breakdown of Classical Solutions 109
4.3 The Cauchy Problem: Weak Solutions 112
4.4 Nonuniqueness of Weak Solutions 113
4.5 Entropy Admissibility Condition 114
4.6 The Vanishing Viscosity Approach 118
4.7 Initial-Boundary Value Problems 122
4.8 Notes 126
V Entropy and the Stability of Classical Solutions 128
5.1 Convex Entropy and the Existence of Classical Solutions 129
5.2 The Role of Damping and Relaxation 139
5.3 Convex Entropy and the Stability of Classical Solutions 147
5.4 Involutions 150
5.5 Contingent Entropies and Polyconvexity 160
5.6 Initial-Boundary Value Problems 169
5.7 Notes 172
VI The L1 Theory for Scalar Conservation Laws 176
6.1 The Cauchy Problem: Perseverance and Demiseof Classical Solutions 177
6.2 Admissible Weak Solutions and their Stability Properties 179
6.3 The Method of Vanishing Viscosity 184
6.4 Solutions as Trajectories of a Contraction Semigroup 189
6.5 The Layering Method 195
6.6 Relaxation 198
6.7 A Kinetic Formulation 205
6.8 Fine Structure of L Solutions 211
6.9 Initial-Boundary Value Problems 214
6.10 The L1 Theory for Systems of Conservation Laws 219
6.11 Notes 223
VII Hyperbolic Systems of Balance Laws in One-Space Dimension 226
7.1 Balance Laws in One-Space Dimension 226
7.2 Hyperbolicity and Strict Hyperbolicity 234
7.3 Riemann Invariants 237
7.4 Entropy-Entropy Flux Pairs 242
7.5 Genuine Nonlinearity and Linear Degeneracy 245
7.6 Simple Waves 247
7.7 Explosion of Weak Fronts 251
7.8 Existence and Breakdown of Classical Solutions 252
7.9 Weak Solutions 256
7.10 Notes 257
VIII Admissible Shocks 261
8.1 Strong Shocks, Weak Shocks, and Shocks of Moderate Strength 261
8.2 The Hugoniot Locus 264
8.3 The Lax Shock Admissibility Criterion Compressive, Overcompressive and Undercompressive Shocks
8.4 The Liu Shock Admissibility Criterion 276
8.5 The Entropy Shock Admissibility Criterion 278
8.6 Viscous Shock Profiles 282
8.7 Nonconservative Shocks 294
8.8 Notes 295
IX Admissible Wave Fans and the Riemann Problem 300
9.1 Self-Similar Solutions and the Riemann Problem 300
9.2 Wave Fan Admissibility Criteria 303
9.3 Solution of the Riemann Problem via Wave Curves 304
9.4 Systems with Genuinely Nonlinear or Linearly Degenerate Characteristic Families 307
9.5 General Strictly Hyperbolic Systems 312
9.6 Failure of Existence or Uniqueness Delta Shocks and Transitional Waves
9.7 The Entropy Rate Admissibility Criterion 319
9.8 Viscous Wave Fans 328
9.9 Interaction of Wave Fans 338
9.10 Breakdown of Weak Solutions 346
9.11 Notes 349
X Generalized Characteristics 354
10.1 BV Solutions 354
10.2 Generalized Characteristics 355
10.3 Extremal Backward Characteristics 357
10.4 Notes 359
XI Genuinely Nonlinear Scalar Conservation Laws 360
11.1 Admissible BV Solutions and Generalized Characteristics 361
11.2 The Spreading of Rarefaction Waves 364
11.3 Regularity of Solutions 365
11.4 Divides, Invariants and the Lax Formula 369
11.5 Decay of Solutions Induced by Entropy Dissipation 373
11.6 Spreading of Characteristics and Development of N-Waves 375
11.7 Confinement of Characteristicsand Formation of Saw-toothed Profiles 377
11.8 Comparison Theorems and L1 Stability 379
11.9 Genuinely Nonlinear Scalar Balance Laws 387
11.10 Balance Laws with Linear Excitation 391
11.11 An Inhomogeneous Conservation Law 394
11.12 Notes 399
XII Genuinely Nonlinear Systems of Two Conservation Laws 402
12.1 Notation and Assumptions 402
12.2 Entropy-Entropy Flux Pairs and the Hodograph Transformation 404
12.3 Local Structure of Solutions 407
12.4 Propagation of Riemann Invariants Along Extremal Backward Characteristics 410
12.5 Bounds on Solutions 427
12.6 Spreading of Rarefaction Waves 439
12.7 Regularity of Solutions 444
12.8 Initial Data in L1 446
12.9 Initial Data with Compact Support 450
12.10 Periodic Solutions 456
12.11 Notes 461
XIII The Random Choice Method 463
13.1 The Construction Scheme 463
13.2 Compactness and Consistency 466
13.3 Wave Interactions, Approximate Conservation Laws and Approximate Characteristics in Genuinely Nonlinear Systems 472
13.4 The Glimm Functional for Genuinely Nonlinear Systems 476
13.5 Bounds on the Total Variation for Genuinely Nonlinear Systems 481
13.6 Bounds on the Supremum for Genuinely Nonlinear Systems 483
13.7 General Systems 485
13.8 Wave Tracing 488
13.9 Inhomogeneous Systems of Balance Laws 491
13.10 Notes 502
XIV The Front Tracking Method and Standard Riemann Semigroups 504
14.1 Front Tracking for Scalar Conservation Laws 505
14.2 Front Tracking for Genuinely Nonlinear Systems of Conservation Laws 507
14.3 The Global Wave Pattern 512
14.4 Approximate Solutions 513
14.5 Bounds on the Total Variation 515
14.6 Bounds on the Combined Strength of Pseudoshocks 518
14.7 Compactness and Consistency 521
14.8 Continuous Dependence on Initial Data 523
14.9 The Standard Riemann Semigroup 527
14.10 Uniqueness of Solutions 528
14.11 Continuous Glimm Functionals,Spreading of Rarefaction Waves,and Structure of Solutions 534
14.12 Stability of Strong Waves 537
14.13 Notes 539
XV Construction of BV Solutions by the Vanishing Viscosity Method 543
15.1 The Main Result 543
15.2 Road Map to the Proof of Theorem 15.1.1 545
15.3 The Effects of Diffusion 547
15.4 Decomposition into Viscous Traveling Waves 550
15.5 Transversal Wave Interactions 554
15.6 Interaction of Waves of the Same Family 558
15.7 Energy Estimates 562
15.8 Stability Estimates 565
15.9 Notes 568
XVI Compensated Compactness 570
16.1 The Young Measure 571
16.2 Compensated Compactness and the div-curl Lemma 572
16.3 Measure-Valued Solutions for Systems of Conservation Laws and Compensated Compactness 573
16.4 Scalar Conservation Laws 576
16.5 A Relaxation Scheme for Scalar Conservation Laws 578
16.6 Genuinely Nonlinear Systems of Two Conservation Laws 581
16.7 The System of Isentropic Elasticity 584
16.8 The System of Isentropic Gas Dynamics 589
16.9 Notes 592
XVII Conservation Laws in Two Space Dimensions 597
17.1 Self-Similar Solutions for Multidimensional Scalar Conservation Laws 597
17.2 Steady Planar Isentropic Gas Flow 600
17.3 Self-Similar Planar Irrotational Isentropic Gas Flow 604
17.4 Supersonic Isentropic Gas Flow Past a Ramp of Gentle Slope 607
17.5 Regular Shock Reflection on a Wall 612
17.6 Shock Collision with a Steep Ramp 615
17.7 Notes 618
Bibliography 621
Author Index 716
Subject Index 725