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
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).
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
II Introduction to Continuum Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
IV The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.1 The Cauchy Problem: Classical Solutions . . . . . . . . . . . . . . . . . . . 77
4.2 Breakdown of Classical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3 The Cauchy Problem: Weak Solutions . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Nonuniqueness of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5 Entropy Admissibility Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.6 The Vanishing Viscosity Approach . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.7 Initial-Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.8 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
V Entropy and the Stability of Classical Solutions . . . . . . . . . . . . . . . . . 111
5.1 Convex Entropy and the Existence of Classical Solutions . . . . . . 112
5.2 Relative Entropy and the Stability of Classical Solutions . . . . . . 122
5.3 Involutions and Contingent Entropies . . . . . . . . . . . . . . . . . . . . . . . 125
5.4 Contingent Entropies and Polyconvexity . . . . . . . . . . . . . . . . . . . . 138
5.5 The Role of Damping and Relaxation . . . . . . . . . . . . . . . . . . . . . . 146
5.6 Initial-Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
VI The L1 Theory for Scalar Conservation Laws . . . . . . . . . . . . . . . . . . . 175
6.1 The Cauchy Problem: Perseverance and Demise
of Classical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.2 Admissible Weak Solutions and their Stability Properties . . . . . . 178
6.3 The Method of Vanishing Viscosity . . . . . . . . . . . . . . . . . . . . . . . . 183
6.4 Solutions as Trajectories of a Contraction Semigroup and the
Large Time Behavior of Periodic Solutions . . . . . . . . . . . . . . . . . . 188
6.5 The Layering Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.6 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
6.7 A Kinetic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
6.8 Fine Structure of L¥ Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
6.9 Initial-Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 215
6.10 The L1 Theory for Systems of Conservation Laws . . . . . . . . . . . . 220
6.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
VII Hyperbolic Systems of Balance Laws in One-Space Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
7.1 Balance Laws in One-Space Dimension . . . . . . . . . . . . . . . . . . . . 227
7.2 Hyperbolicity and Strict Hyperbolicity . . . . . . . . . . . . . . . . . . . . . 235
7.3 Riemann Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
7.4 Entropy-Entropy Flux Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
7.5 Genuine Nonlinearity and Linear Degeneracy . . . . . . . . . . . . . . . . 245
7.6 Simple Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
7.7 Explosion of Weak Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
7.8 Existence and Breakdown of Classical Solutions . . . . . . . . . . . . . 253
7.9 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
7.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
VIII Admissible Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
8.1 Strong Shocks, Weak Shocks, and Shocks of Moderate Strength 263
8.2 The Hugoniot Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
Compressive, Overcompressive and Undercompressive Shocks . 272
8.4 The Liu Shock Admissibility Criterion . . . . . . . . . . . . . . . . . . . . . 278
8.5 The Entropy Shock Admissibility Criterion . . . . . . . . . . . . . . . . . . 280
8.6 Viscous Shock Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
8.7 Nonconservative Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
8.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
IX Admissible Wave Fans and the Riemann Problem. . . . . . . . . . . . . . . . 303
9.1 Self-Similar Solutions and the Riemann Problem . . . . . . . . . . . . . 303
9.2 Wave Fan Admissibility Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
9.3 Solution of the Riemann Problem via Wave Curves . . . . . . . . . . . 309
9.4 Systems with Genuinely Nonlinear
or Linearly Degenerate Characteristic Families . . . . . . . . . . . . . . . 312
9.5 General Strictly Hyperbolic Systems . . . . . . . . . . . . . . . . . . . . . . . 316
9.6 Failure of Existence or Uniqueness;
Delta Shocks and Transitional Waves . . . . . . . . . . . . . . . . . . . . . . . 320^lt;9.7 The Entropy Rate Admissibility Criterion . . . . . . . . . . . . . . . . . . . 323
9.8 Viscous Wave Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
9.9 Interaction of Wave Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
9.10 Breakdown of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
9.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
X Generalized Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
10.1 BV Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
10.2 Generalized Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
10.3 Extremal Backward Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 362
10.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
XI Scalar Conservation Laws in One Space Dimension . . . . . . . . . . . . . 367
11.1 Admissible BV Solutions and Generalized Characteristics . . . . . 368
11.2 The Spreading of Rarefaction Waves . . . . . . . . . . . . . . . . . . . . . . . 371
11.3 Regularity of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
11.4 Divides, Invariants and the Lax Formula . . . . . . . . . . . . . . . . . . . . 377
11.5 Decay of Solutions Induced by Entropy Dissipation . . . . . . . . . . 380
11.6 Spreading of Characteristics and Development of N-Waves . . . . 383
11.7 Confinement of Characteristics
and Formation of Saw-toothed Profiles . . . . . . . . . . . . . . . . . . . . . 384
11.8 Comparison Theorems and L1 Stability . . . . . . . . . . . . . . . . . . . . . 386
11.9 Genuinely Nonlinear Scalar Balance Laws . . . . . . . . . . . . . . . . . . 395
11.10 Balance Laws with Linear Excitation . . . . . . . . . . . . . . . . . . . . . . . 399
11.11 An Inhomogeneous Conservation Law. . . . . . . . . . . . . . . . . . . . . . 401
11.12 When Genuine Nonlinearity Fails . . . . . . . . . . . . . . . . . . . . . . . . . . 406
11.13 Entropy Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
11.14 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
XII Genuinely Nonlinear Systems of Two Conservation Laws . . . . . . . . . 427
12.1 Notation and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
12.2 Entropy-Entropy Flux Pairs and the Hodograph Transformation 429
12.3 Local Structure of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
12.4 Propagation of Riemann Invariants
Along Extremal Backward Characteristics . . . . . . . . . . . . . . . . . . 435
12.5 Bounds on Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452
12.6 Spreading of Rarefaction Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
12.7 Regularity of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
12.8 Initial Data in L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
12.9 Initial Data with Compact Support . . . . . . . . . . . . . . . . . . . . . . . . . 475
12.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
XIII The Random Choice Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
13.1 The Construction Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
13.2 Compactness and Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
13.3 Wave Interactions in Genuinely Nonlinear Systems . . . . . . . . . . . 498
13.4 The Glimm Functional for Genuinely Nonlinear Systems . . . . . . 500
13.5 Bounds on the Total Variation
for Genuinely Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 505
13.6 Bounds on the Supremum for Genuinely Nonlinear Systems . . . 507
13.7 General Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
13.8 Wave Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
13.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
XIV The Front Tracking Method and Standard Riemann Semigroups . . 517
14.1 Front Tracking for Scalar Conservation Laws . . . . . . . . . . . . . . . . 518
14.2 Front Tracking for Genuinely Nonlinear
Systems of Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
14.3 The Global Wave Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
14.4 Approximate Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
14.5 Bounds on the Total Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
14.6 Bounds on the Combined Strength of Pseudoshocks . . . . . . . . . . 531
14.7 Compactness and Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
14.8 Continuous Dependence on Initial Data . . . . . . . . . . . . . . . . . . . . . 536
14.9 The Standard Riemann Semigroup . . . . . . . . . . . . . . . . . . . . . . . . . 540
14.10 Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
14.11 Continuous Glimm Functionals,
Spreading of Rarefaction Waves,
and Structure of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
14.12 Stability of Strong Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
14.13 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
XV Construction of BV Solutions by the Vanishing Viscosity Method . . 557
15.1 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
15.2 Road Map to the Proof of Theorem 15.1.1 . . . . . . . . . . . . . . . . . . . 559
15.3 The Effects of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
15.4 Decomposition into Viscous Traveling Waves . . . . . . . . . . . . . . . . 564
15.5 Transversal Wave Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568
15.6 Interaction of Waves of the Same Family . . . . . . . . . . . . . . . . . . . . 572
15.7 Energy Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576
15.8 Stability Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
15.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582
XVI BV Solutions for Systems of Balance Laws . . . . . . . . . . . . . . . . . . . . . . 585
16.1 The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
16.2 Strong Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
16.3 Redistribution of Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
16.4 Bounds on the Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
16.5 L1 Stability Via Entropy with Conical Singularity at the Origin . 606
16.6 L1 Stability when the Source is Partially Dissipative . . . . . . . . . . 609
16.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622
XVII Compensated Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
17.1 The Young Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624
17.2 Compensated Compactness and the div-curl Lemma . . . . . . . . . . 625
17.3 Measure-Valued Solutions for Systems of Conservation Laws
and Compensated Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . 626
17.4 Scalar Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
17.5 A Relaxation Scheme for Scalar Conservation Laws . . . . . . . . . . 631
17.6 Genuinely Nonlinear Systems of Two Conservation Laws . . . . . 634
17.7 The System of Isentropic Elasticity . . . . . . . . . . . . . . . . . . . . . . . . 637
17.8 The System of Isentropic Gas Dynamics . . . . . . . . . . . . . . . . . . . . 64217.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648
XVIII Steady and Self-similar Solutions in Multi-Space Dimensions . . . . . 655
18.1 Self-Similar Solutions for Multidimensional Scalar
Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655
18.2 Steady Planar Isentropic Gas Flow . . . . . . . . . . . . . . . . . . . . . . . . . 65818.3 Self-Similar Planar Irrotational Isentropic Gas Flow . . . . . . . . . . 663
18.4 Supersonic Isentropic Gas Flow Past a Ramp . . . . . . . . . . . . . . . . 667
18.5 Regular Shock Reflection on a Wall . . . . . . . . . . . . . . . . . . . . . . . . 672
18.6 Shock Collision with a Ramp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
18.7 Isometric Immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678
18.8 Cavitation in Elastodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682
18.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823
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