Hyperbolic Conservation Laws in Continuum Physics (Grundlehren der mathematischen Wissenschaften Book 325)

This is a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws. It illustrates the essential role of continuum thermodynamics in providing motivation and direction for the development of the mathematical theory while also serving as the principal source of applications. The reader is expected to have a certain mathematical sophistication and to be familiar with (at least) the rudiments of analysis and the qualitative theory of partial differential equations, whereas prior exposure to continuum physics is not required. The target group of readers would consist of (a) experts in the mathematical theory of hyperbolic systems of conservation laws who wish to learn about the connection with classical physics; (b) specialists in continuum mechanics who may need analytical tools; (c) experts in numerical analysis who wish to learn the underlying mathematical theory; and (d) analysts and graduate students who seek introduction to the theory of hyperbolic systems of conservation laws. This new edition places increased emphasis on hyperbolic systems of balance laws with dissipative source, modeling relaxation phenomena. It also presents an account of recent developments on the Euler equations of compressible gas dynamics. Furthermore, the presentation of a number of topics in the previous edition has been revised, expanded and brought up to date, and has been enriched with new applications to elasticity and differential geometry. The bibliography, also expanded and updated, now comprises close to two thousand titles. From the reviews of the 3rd edition: "This is the third edition of the famous book by C.M. Dafermos. His masterly written book is, surely, the most complete exposition in the subject." Evgeniy Panov, Zentralblatt MATH "A monumental book encompassing all aspects of the mathematical theory of hyperbolic conservation laws, widely recognized as the "Bible" on the subject." Philippe G. LeFloch, Math. Reviews.--Résumé de l'éditeur Preface to the Fourth Edition 7 Acknowledgments 9 Introduction 10 A Sketch of the Early History of Hyperbolic Conservation Laws 15 Contents 31 I Balance Laws 37 1.1 Formulation of the Balance Law 38 1.2 Reduction to Field Equations 39 1.3 Change of Coordinates and a Trace Theorem 43 1.4 Systems of Balance Laws 48 1.5 Companion Balance Laws 49 1.6 Weak and Shock Fronts 51 1.7 Survey of the Theory of BV Functions 53 1.8 BV Solutions of Systems of Balance Laws 57 1.9 Rapid Oscillations and the Stabilizing Effect of Companion Balance Laws 59 1.10 Notes 59 II Introduction to Continuum Physics 61 2.1 Kinematics 61 2.2 Balance Laws in Continuum Physics 64 2.3 The Balance Laws of Continuum Thermomechanics 67 2.4 Material Frame Indifference 71 2.5 Thermoelasticity 72 2.6 Thermoviscoelasticity 80 2.7 Incompressibility 83 2.8 Relaxation 84 2.9 Notes 85 III Hyperbolic Systems of Balance Laws 88 3.1 Hyperbolicity 88 3.2 Entropy-Entropy Flux Pairs 89 3.3 Examples of Hyperbolic Systems of Balance Laws 91 3.4 Notes 108 IV The Cauchy Problem 111 4.1 The Cauchy Problem: Classical Solutions 111 4.2 Breakdown of Classical Solutions 114 4.3 The Cauchy Problem: Weak Solutions 116 4.4 Nonuniqueness of Weak Solutions 117 4.5 Entropy Admissibility Condition 118 4.6 The Vanishing Viscosity Approach 124 4.7 Initial-Boundary Value Problems 128 4.8 Euler Equations 131 4.9 Notes 141 V Entropy and the Stability of Classical Solutions 144 5.1 Convex Entropy and the Existence of Classical Solutions 145 5.2 Relative Entropy and the Stability of Classical Solutions 155 5.3 Involutions and Contingent Entropies 158 5.4 Contingent Entropies and Polyconvexity 171 5.5 The Role of Damping and Relaxation 179 5.6 Initial-Boundary Value Problems 193 5.7 Notes 203 VI The L1 Theory for Scalar Conservation Laws 208 6.1 The Cauchy Problem: Perseverance and Demise of Classical Solutions 209 6.2 AdmissibleWeak Solutions and their Stability Properties 211 6.3 The Method of Vanishing Viscosity 216 6.4 Solutions as Trajectories of a Contraction Semigroup and the Large Time Behavior of Periodic Solutions 221 6.5 The Layering Method 228 6.6 Relaxation 232 6.7 A Kinetic Formulation 238 6.8 Fine Structure of L∞ Solutions 245 6.9 Initial-Boundary Value Problems 248 6.10 The L1 Theory for Systems of Conservation Laws 253 6.11 Notes 256 VII Hyperbolic Systems of Balance Laws in One-Space Dimension 260 7.1 Balance Laws in One-Space Dimension 260 7.2 Hyperbolicity and Strict Hyperbolicity 268 7.3 Riemann Invariants 271 7.4 Entropy-Entropy Flux Pairs 276 7.5 Genuine Nonlinearity and Linear Degeneracy 278 7.6 SimpleWaves 280 7.7 Explosion ofWeak Fronts 285 7.8 Existence and Breakdown of Classical Solutions 286 7.9 Weak Solutions 290 7.10 Notes 291 VIII Admissible Shocks 295 8.1 Strong Shocks,Weak Shocks, and Shocks of Moderate Strength 295 8.2 The Hugoniot Locus 298 8.3 The Lax Shock Admissibility Criterion; Compressive, Overcompressive and Undercompressive Shocks 304 8.4 The Liu Shock Admissibility Criterion 310 8.5 The Entropy Shock Admissibility Criterion 312 8.6 Viscous Shock Profiles 317 8.7 Nonconservative Shocks 328 8.8 Notes 329 IX Admissible Wave Fans and the Riemann Problem 335 9.1 Self-Similar Solutions and the Riemann Problem 335 9.2 Wave Fan Admissibility Criteria 339 9.3 Solution of the Riemann Problem viaWave Curves 341 9.4 Systems with Genuinely Nonlinear or Linearly Degenerate Characteristic Families 344 9.5 General Strictly Hyperbolic Systems 348 9.6 Failure of Existence or Uniqueness; Delta Shocks and TransitionalWaves 352 9.7 The Entropy Rate Admissibility Criterion 355 9.8 ViscousWave Fans 364 9.9 Interaction ofWave Fans 375 9.10 Breakdown ofWeak Solutions 382 9.11 Notes 386 X Generalized Characteristics 391 10.1 BV Solutions 391 10.2 Generalized Characteristics 392 10.3 Extremal Backward Characteristics 394 10.4 Notes 397 XI Scalar Conservation Laws in One Space Dimension 398 11.1 Admissible BV Solutions and Generalized Characteristics 399 11.2 The Spreading of RarefactionWaves 402 11.3 Regularity of Solutions 403 11.4 Divides, Invariants and the Lax Formula 408 11.5 Decay of Solutions Induced by Entropy Dissipation 411 11.6 Spreading of Characteristics and Development of N-Waves 414 11.7 Confinement of Characteristics and Formation of Saw-toothed Profiles 415 11.8 Comparison Theorems and L1 Stability 417 11.9 Genuinely Nonlinear Scalar Balance Laws 426 11.10 Balance Laws with Linear Excitation 430 11.11 An Inhomogeneous Conservation Law 432 11.12 When Genuine Nonlinearity Fails 437 11.13 Entropy Production 449 11.14 Notes 453 XII Genuinely Nonlinear Systems of Two Conservation Laws 458 12.1 Notation and Assumptions 458 12.2 Entropy-Entropy Flux Pairs and the Hodograph Transformation 460 12.3 Local Structure of Solutions 463 12.4 Propagation of Riemann Invariants Along Extremal Backward Characteristics 466 12.5 Bounds on Solutions 483 12.6 Spreading of RarefactionWaves 495 12.7 Regularity of Solutions 500 12.8 Initial Data in L1 502 12.9 Initial Data with Compact Support 506 12.10 Periodic Solutions 512 12.11 Notes 517 XIII The Random Choice Method 519 13.1 The Construction Scheme 519 13.2 Compactness and Consistency 522 13.3 Wave Interactions in Genuinely Nonlinear Systems 528 13.4 The Glimm Functional for Genuinely Nonlinear Systems 530 13.5 Bounds on the Total Variation for Genuinely Nonlinear Systems 535 13.6 Bounds on the Supremum for Genuinely Nonlinear Systems 537 13.7 General Systems 539 13.8 Wave Tracing 542 13.9 Notes 545 XIV The Front Tracking Method and Standard Riemann Semigroups 547 14.1 Front Tracking for Scalar Conservation Laws 548 14.2 Front Tracking for Genuinely Nonlinear Systems of Conservation Laws 550 14.3 The GlobalWave Pattern 555 14.4 Approximate Solutions 556 14.5 Bounds on the Total Variation 558 14.6 Bounds on the Combined Strength of Pseudoshocks 561 14.7 Compactness and Consistency 564 14.8 Continuous Dependence on Initial Data 566 14.9 The Standard Riemann Semigroup 570 14.10 Uniqueness of Solutions 571 14.11 Continuous Glimm Functionals, Spreading of RarefactionWaves, and Structure of Solutions 577 14.12 Stability of StrongWaves 580 14.13 Notes 582 XV Construction of BV Solutions by the Vanishing Viscosity Method 586 15.1 The Main Result 586 15.2 Road Map to the Proof of Theorem 15.1.1 588 15.3 The Effects of Diffusion 590 15.4 Decomposition into Viscous TravelingWaves 593 15.5 TransversalWave Interactions 597 15.6 Interaction ofWaves of the Same Family 601 15.7 Energy Estimates 605 15.8 Stability Estimates 608 15.9 Notes 611 XVI BV Solutions for Systems of Balance Laws 613 16.1 The Cauchy Problem 614 16.2 Strong Dissipation 617 16.3 Redistribution of Damping 621 16.4 Bounds on the Variation 623 16.5 L1 Stability Via Entropy with Conical Singularity at the Origin 634 16.6 L1 Stability when the Source is Partially Dissipative 637 16.7 Notes 650 XVII Compensated Compactness 651 17.1 The Young Measure 652 17.2 Compensated Compactness and the div-curl Lemma 653 17.3 Measure-Valued Solutions for Systems of Conservation Laws and Compensated Compactness 654 17.4 Scalar Conservation Laws 657 17.5 A Relaxation Scheme for Scalar Conservation Laws 659 17.6 Genuinely Nonlinear Systems of Two Conservation Laws 662 17.7 The System of Isentropic Elasticity 665 17.8 The System of Isentropic Gas Dynamics 670 17.9 Notes 676 XVIII Steady and Self-similar Solutions in Multi-Space Dimensions 682 18.1 Self-Similar Solutions for Multidimensional Scalar Conservation Laws 682 18.2 Steady Planar Isentropic Gas Flow 685 18.3 Self-Similar Planar Irrotational Isentropic Gas Flow 690 18.4 Supersonic Isentropic Gas Flow Past a Ramp 694 18.5 Regular Shock Reflection on aWall 699 18.6 Shock Collision with a Ramp 702 18.7 Isometric Immersions 705 18.8 Cavitation in Elastodynamics 709 18.9 Notes 713 Bibliography 717 Author Index 837 Subject Index 847 OLD TEXT 4th Edition to be replaced: This is a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws. It illustrates the essential role of continuum thermodynamics in providing motivation and direction for the development of the mathematical theory while also serving as the principal source of applications. The reader is expected to have a certain mathematical sophistication and to be familiar with (at least) the rudiments of analysis and the qualitative theory of partial differential equations, whereas prior exposure to continuum physics is not required. The target group of readers would consist of (a) experts in the mathematical theory of hyperbolic systems of conservation laws who wish to learn about the connection with classical physics; (b) specialists in continuum mechanics who may need analytical tools; (c) experts in numerical analysis who wish to learn the underlying mathematical theory; and (d) analysts and graduate students who seek introduction to the theory of hyperbolic systems of conservation laws.This new edition places increased emphasis on hyperbolic systems of balance laws with dissipative source, modeling relaxation phenomena. It also presents an account of recent developments on the Euler equations of compressible gas dynamics. Furthermore, the presentation of a number of topics in the previous edition has been revised, expanded and brought up to date, and has been enriched with new applications to elasticity and differential geometry. The bibliography, also expanded and updated, now comprises close to two thousand titles.From the reviews of the 3rd edition:'This is the third edition of the famous book by C.M. Dafermos. His masterly written book is, surely, the most complete exposition in the subject.'Evgeniy Panov, Zentralblatt MATH'A monumental book encompassing all aspects of the mathematical theory of hyperbolic conservation laws, widely recognized as the'Bible'on the subject.'Philippe G. LeFloch, Math. Reviews This is a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws. It illustrates the essential role of continuum thermodynamics in providing motivation and direction for the development of the mathematical theory while also serving as the principal source of applications. The reader is expected to have a certain mathematical sophistication and to be familiar with (at least) the rudiments of analysis and the qualitative theory of partial differential equations, whereas prior exposure to continuum physics is not required. The target group of readers would consist of (a) experts in the mathematical theory of hyperbolic systems of conservation laws who wish to learn about the connection with classical physics; (b) specialists in continuum mechanics who may need analytical tools; (c) experts in numerical analysis who wish to learn the underlying mathematical theory; and (d) analysts and graduate students who seek introduction to the theory of hyperbolic systems of conservation laws. This new edition places increased emphasis on hyperbolic systems of balance laws with dissipative source, modeling relaxation phenomena. It also presents an account of recent developments on the Euler equations of compressible gas dynamics. Furthermore, the presentation of a number of topics in the previous edition has been revised, expanded and brought up to date, and has been enriched with new applications to elasticity and differential geometry. The bibliography, also expanded and updated, now comprises close to two thousand titles. From the reviews of the 3rd edition: "This is the third edition of the famous book by C.M. Dafermos. His masterly written book is, surely, the most complete exposition in the subject." Evgeniy Panov, Zentralblatt MATH "A monumental book encompassing all aspects of the mathematical theory of hyperbolic conservation laws, widely recognized as the "Bible" on the subject." Philippe G. LeFloch, Math. Reviews.--Résumé de l'éditeur Front Matter....Pages I-XXXVIII Balance Laws....Pages 1-24 Introduction to Continuum Physics....Pages 25-51 Hyperbolic Systems of Balance Laws....Pages 53-75 The Cauchy Problem....Pages 77-109 Entropy and the Stability of Classical Solutions....Pages 111-174 The \(L^1\) Theory for Scalar Conservation Laws....Pages 175-226 Hyperbolic Systems of Balance Laws in One-Space Dimension....Pages 227-261 Admissible Shocks....Pages 263-302 Admissible Wave Fans and the Riemann Problem....Pages 303-358 Generalized Characteristics....Pages 359-365 Scalar Conservation Laws in One Space Dimension....Pages 367-426 Genuinely Nonlinear Systems of Two Conservation Laws....Pages 427-487 The Random Choice Method....Pages 489-516 The Front Tracking Method and Standard Riemann Semigroups....Pages 517-555 Construction of BV Solutions by the Vanishing Viscosity Method....Pages 557-583 BV Solutions for Systems of Balance Laws....Pages 585-622 Compensated Compactness....Pages 623-653 Steady and Self-Similar Solutions in Multi-Space Dimensions....Pages 655-689 Back Matter....Pages 691-826 This masterly exposition of the mathematical theory of hyperbolic system laws brings out the intimate connection with continuum thermodynamics, emphasizing issues in which the analysis may reveal something about the physics and, in return, the underlying physical structure may direct and drive the analysis. The reader should have a certain mathematical sophistication and be familiar with (at least) the rudiments of the qualitative theory of PDE, whereas the required notions from continuum physics are introduced from scratch. The target group of readers would consist of (a) experts in the mathematical theory of hyperbolic systems of conservation laws who wish to learn about the connection with classical physics; (b) specialists in continuum mechanics; (c) experts in numerical analysis who wish to learn the underlying mathematical theory; and (d) analysts and graduate students who seek introduction to the theory of hyperbolic systems of conservation laws."