Hyperbolic conservation laws in continuum physics:
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| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2005
|
| Edition: | 2. ed. |
| Series: | Grundlehren der mathematischen Wissenschaften
325 |
| Subjects: | |
| Online Access: | DE-634 DE-91 DE-384 DE-355 DE-703 DE-739 Volltext Beschreibung für den Leser Inhaltsverzeichnis |
| Physical Description: | 1 Online-Ressource (XIX, 626 S.) Ill., graph. Darst. |
| ISBN: | 3540254528 9783540254522 9783540290896 |
| DOI: | 10.1007/3-540-29089-3 |
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Record in the Search Index
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| adam_text |
Contents
I Balance Laws 1
1.1 Formulation of the Balance Law 2
1.2 Reduction to Field Equations 2
1.3 Change of Coordinates and a Trace Theorem 6
1.4 Systems of Balance Laws 11
1.5 Companion Balance Laws 12
1.6 Weak and Shock Fronts 15
1.7 Survey of the Theory of BV Functions 16
1.8 BV Solutions of Systems of Balance Laws 20
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 43
2.7 Incompressibility 46
2.8 Relaxation 47
2.9 Notes 48
III Hyperbolic Systems of Balance Laws 51
3.1 Hyperbolicity 51
3.2 Entropy Entropy Flux Pairs 52
3.3 Examples of Hyperbolic Systems of Balance Laws 54
3.4 Notes 64
XVI Contents
IV The Cauchy Problem 67
4.1 The Cauchy Problem: Classical Solutions 67
4.2 Breakdown of Classical Solutions 70
4.3 The Cauchy Problem: Weak Solutions 73
4.4 Nonuniqueness of Weak Solutions 74
4.5 Entropy Admissibility Condition 75
4.6 The Vanishing Viscosity Approach 78
4.7 Initial Boundary Value Problems 82
4.8 Notes 85
V Entropy and the Stability of Classical Solutions 87
5.1 Convex Entropy and the Existence of Classical Solutions 88
5.2 The Role of Damping and Relaxation 92
5.3 Convex Entropy and the Stability of Classical Solutions 98
5.4 Involutions 101
5.5 Contingent Entropies and Polyconvexity Ill
5.6 Notes 120
VI The L1 Theory for Scalar Conservation Laws 123
6.1 The Cauchy Problem: Perseverance and Demise
of Classical Solutions 124
6.2 Admissible Weak Solutions and their Stability Properties 126
6.3 The Method of Vanishing Viscosity 131
6.4 Solutions as Trajectories of a Contraction Semigroup 136
6.5 The Layering Method 142
6.6 Relaxation 146
6.7 A Kinetic Formulation 153
6.8 Fine Structure of L°° Solutions 159
6.9 Initial Boundary Value Problems 162
6.10 The Lx Theory for Systems of Conservation Laws 166
6.11 Notes 170
VII Hyperbolic Systems of Balance Laws in One Space Dimension . 173
7.1 Balance Laws in One Space Dimension 173
7.2 Hyperbolicity and Strict Hyperbolicity 180
7.3 Riemann Invariants 183
7.4 Entropy Entropy Flux Pairs 188
7.5 Genuine Nonlinearity and Linear Degeneracy 190
7.6 Simple Waves 192
7.7 Explosion of Weak Fronts 197
7.8 Breakdown of Classical Solutions 198
7.9 Weak Solutions 201
7.10 Notes 202
Contents XVII
VIII Admissible Shocks 205
8.1 Strong Shocks, Weak Shocks, and Shocks of Moderate Strength 205
8.2 The Hugoniot Locus 207
8.3 The Lax Shock Admissibility Criterion;
Compressive, Overcompressive and Undercompressive Shocks . 213
8.4 The Liu Shock Admissibility Criterion 218
8.5 The Entropy Shock Admissibility Criterion 220
8.6 Viscous Shock Profiles 224
8.7 Nonconservative Shocks 234
8.8 Notes 235
IX Admissible Wave Fans and the Riemann Problem 239
9.1 Self similar Solutions and the Riemann Problem 239
9.2 Wave Fan Admissibility Criteria 242
9.3 Solution of the Riemann Problem via Wave Fan Curves 243
9.4 Systems with Genuinely Nonlinear
or Linearly Degenerate Characteristic Families 246
9.5 General Strictly Hyperbolic Systems 248
9.6 Failure of Existence or Uniqueness;
Delta Shocks and Transitional Waves 253
9.7 The Entropy Rate Admissibility Criterion 256
9.8 Viscous Wave Fans 263
9.9 Interaction of Wave Fans 274
9.10 Breakdown of Weak Solutions 281
9.11 Self similar Solutions for Multidimensional
Conservation Laws 285
9.12 Notes 289
X Generalized Characteristics 295
10.1 BV Solutions 295
10.2 Generalized Characteristics 296
10.3 Extremal Backward Characteristics 298
10.4 Notes 300
XI Genuinely Nonlinear Scalar Conservation Laws 301
11.1 Admissible B V Solutions and Generalized Characteristics 302
11.2 The Spreading of Rarefaction Waves 305
11.3 Regularity of Solutions 306
11.4 Divides, Invariants and the Lax Formula 311
11.5 Decay of Solutions Induced by Entropy Dissipation 314
11.6 Spreading of Characteristics and Development of A? Waves 316
11.7 Confinement of Characteristics
and Formation of Saw toothed Profiles 318
11.8 Comparison Theorems and L1 Stability 320
11.9 Genuinely Nonlinear Scalar Balance Laws 328
XVIII Contents
11.10 Balance Laws with Linear Excitation 332
11.11 An Inhomogeneous Conservation Law 335
11.12 Notes 340
XII Genuinely Nonlinear Systems of Two Conservation Laws 343
12.1 Notation and Assumptions 343
12.2 Entropy Entropy Flux Pairs and the Hodograph Transformation 345
12.3 Local Structure of Solutions 348
12.4 Propagation of Riemann Invariants
Along Extremal Backward Characteristics 351
12.5 Bounds on Solutions 368
12.6 Spreading of Rarefaction Waves 381
12.7 Regularity of Solutions 386
12.8 Initial Data in L1 388
12.9 Initial Data with Compact Support 392
12.10 Periodic Solutions 398
12.11 Notes 403
XIII The Random Choice Method 405
13.1 The Construction Scheme 405
13.2 Compactness and Consistency 408
13.3 Wave Interactions, Approximate Conservation Laws
and Approximate Characteristics
in Genuinely Nonlinear Systems 414
13.4 The Glimm Functional for Genuinely Nonlinear Systems 418
13.5 Bounds on the Total Variation
for Genuinely Nonlinear Systems 423
13.6 Bounds on the Supremum for Genuinely Nonlinear Systems . 425
13.7 General Systems 428
13.8 Wave Tracing 430
13.9 Inhomogeneous Systems of Balance Laws 433
13.10 Notes 440
XIV The Front Tracking Method and Standard Riemann Semigroups . 443
14.1 Front Tracking for Scalar Conservation Laws 444
14.2 Front Tracking for Genuinely Nonlinear
Systems of Conservation Laws 447
14.3 The Global Wave Pattern 451
14.4 Approximate Solutions 452
14.5 Bounds on the Total Variation 455
14.6 Bounds on the Combined Strength of Pseudoshocks 458
14.7 Compactness and Consistency 460
14.8 Continuous Dependence on Initial Data 462
14.9 The Standard Riemann Semigroup 466
14.10 Uniqueness of Solutions 468
Contents XIX
14.11 Continuous Glimm Functional,
Spreading of Rarefaction Waves,
and Structure of Solutions 473
14.12 Stability of Strong Waves 476
14.13 Notes 479
XV Construction of BV Solutions by the Vanishing Viscosity Method. 483
15.1 The Main Result 483
15.2 Road Map to the Proof of Theorem 15.1.1 485
15.3 The Effects of Diffusion 487
15.4 Decomposition into Viscous Traveling Waves 490
15.5 Transversal Wave Interactions 494
15.6 Interaction of Waves of the Same Family 498
15.7 Energy Estimates 503
15.8 Stability Estimates 506
15.9 Notes 508
XVI Compensated Compactness 511
16.1 The Young Measure 512
16.2 Compensated Compactness and the div curl Lemma 513
16.3 Measure Valued Solutions for Systems of Conservation Laws
and Compensated Compactness 515
16.4 Scalar Conservation Laws 518
16.5 A Relaxation Scheme for Scalar Conservation Laws 519
16.6 Genuinely Nonlinear Systems of Two Conservation Laws 523
16.7 The System of Isentropic Elasticity 525
16.8 The System of Isentropic Gas Dynamics 530
16.9 Notes 533
Bibliography 537
Author Index 613
Subject Index 621 |
| adam_txt |
Contents
I Balance Laws 1
1.1 Formulation of the Balance Law 2
1.2 Reduction to Field Equations 2
1.3 Change of Coordinates and a Trace Theorem 6
1.4 Systems of Balance Laws 11
1.5 Companion Balance Laws 12
1.6 Weak and Shock Fronts 15
1.7 Survey of the Theory of BV Functions 16
1.8 BV Solutions of Systems of Balance Laws 20
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 43
2.7 Incompressibility 46
2.8 Relaxation 47
2.9 Notes 48
III Hyperbolic Systems of Balance Laws 51
3.1 Hyperbolicity 51
3.2 Entropy Entropy Flux Pairs 52
3.3 Examples of Hyperbolic Systems of Balance Laws 54
3.4 Notes 64
XVI Contents
IV The Cauchy Problem 67
4.1 The Cauchy Problem: Classical Solutions 67
4.2 Breakdown of Classical Solutions 70
4.3 The Cauchy Problem: Weak Solutions 73
4.4 Nonuniqueness of Weak Solutions 74
4.5 Entropy Admissibility Condition 75
4.6 The Vanishing Viscosity Approach 78
4.7 Initial Boundary Value Problems 82
4.8 Notes 85
V Entropy and the Stability of Classical Solutions 87
5.1 Convex Entropy and the Existence of Classical Solutions 88
5.2 The Role of Damping and Relaxation 92
5.3 Convex Entropy and the Stability of Classical Solutions 98
5.4 Involutions 101
5.5 Contingent Entropies and Polyconvexity Ill
5.6 Notes 120
VI The L1 Theory for Scalar Conservation Laws 123
6.1 The Cauchy Problem: Perseverance and Demise
of Classical Solutions 124
6.2 Admissible Weak Solutions and their Stability Properties 126
6.3 The Method of Vanishing Viscosity 131
6.4 Solutions as Trajectories of a Contraction Semigroup 136
6.5 The Layering Method 142
6.6 Relaxation 146
6.7 A Kinetic Formulation 153
6.8 Fine Structure of L°° Solutions 159
6.9 Initial Boundary Value Problems 162
6.10 The Lx Theory for Systems of Conservation Laws 166
6.11 Notes 170
VII Hyperbolic Systems of Balance Laws in One Space Dimension . 173
7.1 Balance Laws in One Space Dimension 173
7.2 Hyperbolicity and Strict Hyperbolicity 180
7.3 Riemann Invariants 183
7.4 Entropy Entropy Flux Pairs 188
7.5 Genuine Nonlinearity and Linear Degeneracy 190
7.6 Simple Waves 192
7.7 Explosion of Weak Fronts 197
7.8 Breakdown of Classical Solutions 198
7.9 Weak Solutions 201
7.10 Notes 202
Contents XVII
VIII Admissible Shocks 205
8.1 Strong Shocks, Weak Shocks, and Shocks of Moderate Strength 205
8.2 The Hugoniot Locus 207
8.3 The Lax Shock Admissibility Criterion;
Compressive, Overcompressive and Undercompressive Shocks . 213
8.4 The Liu Shock Admissibility Criterion 218
8.5 The Entropy Shock Admissibility Criterion 220
8.6 Viscous Shock Profiles 224
8.7 Nonconservative Shocks 234
8.8 Notes 235
IX Admissible Wave Fans and the Riemann Problem 239
9.1 Self similar Solutions and the Riemann Problem 239
9.2 Wave Fan Admissibility Criteria 242
9.3 Solution of the Riemann Problem via Wave Fan Curves 243
9.4 Systems with Genuinely Nonlinear
or Linearly Degenerate Characteristic Families 246
9.5 General Strictly Hyperbolic Systems 248
9.6 Failure of Existence or Uniqueness;
Delta Shocks and Transitional Waves 253
9.7 The Entropy Rate Admissibility Criterion 256
9.8 Viscous Wave Fans 263
9.9 Interaction of Wave Fans 274
9.10 Breakdown of Weak Solutions 281
9.11 Self similar Solutions for Multidimensional
Conservation Laws 285
9.12 Notes 289
X Generalized Characteristics 295
10.1 BV Solutions 295
10.2 Generalized Characteristics 296
10.3 Extremal Backward Characteristics 298
10.4 Notes 300
XI Genuinely Nonlinear Scalar Conservation Laws 301
11.1 Admissible B V Solutions and Generalized Characteristics 302
11.2 The Spreading of Rarefaction Waves 305
11.3 Regularity of Solutions 306
11.4 Divides, Invariants and the Lax Formula 311
11.5 Decay of Solutions Induced by Entropy Dissipation 314
11.6 Spreading of Characteristics and Development of A? Waves 316
11.7 Confinement of Characteristics
and Formation of Saw toothed Profiles 318
11.8 Comparison Theorems and L1 Stability 320
11.9 Genuinely Nonlinear Scalar Balance Laws 328
XVIII Contents
11.10 Balance Laws with Linear Excitation 332
11.11 An Inhomogeneous Conservation Law 335
11.12 Notes 340
XII Genuinely Nonlinear Systems of Two Conservation Laws 343
12.1 Notation and Assumptions 343
12.2 Entropy Entropy Flux Pairs and the Hodograph Transformation 345
12.3 Local Structure of Solutions 348
12.4 Propagation of Riemann Invariants
Along Extremal Backward Characteristics 351
12.5 Bounds on Solutions 368
12.6 Spreading of Rarefaction Waves 381
12.7 Regularity of Solutions 386
12.8 Initial Data in L1 388
12.9 Initial Data with Compact Support 392
12.10 Periodic Solutions 398
12.11 Notes 403
XIII The Random Choice Method 405
13.1 The Construction Scheme 405
13.2 Compactness and Consistency 408
13.3 Wave Interactions, Approximate Conservation Laws
and Approximate Characteristics
in Genuinely Nonlinear Systems 414
13.4 The Glimm Functional for Genuinely Nonlinear Systems 418
13.5 Bounds on the Total Variation
for Genuinely Nonlinear Systems 423
13.6 Bounds on the Supremum for Genuinely Nonlinear Systems . 425
13.7 General Systems 428
13.8 Wave Tracing 430
13.9 Inhomogeneous Systems of Balance Laws 433
13.10 Notes 440
XIV The Front Tracking Method and Standard Riemann Semigroups . 443
14.1 Front Tracking for Scalar Conservation Laws 444
14.2 Front Tracking for Genuinely Nonlinear
Systems of Conservation Laws 447
14.3 The Global Wave Pattern 451
14.4 Approximate Solutions 452
14.5 Bounds on the Total Variation 455
14.6 Bounds on the Combined Strength of Pseudoshocks 458
14.7 Compactness and Consistency 460
14.8 Continuous Dependence on Initial Data 462
14.9 The Standard Riemann Semigroup 466
14.10 Uniqueness of Solutions 468
Contents XIX
14.11 Continuous Glimm Functional,
Spreading of Rarefaction Waves,
and Structure of Solutions 473
14.12 Stability of Strong Waves 476
14.13 Notes 479
XV Construction of BV Solutions by the Vanishing Viscosity Method. 483
15.1 The Main Result 483
15.2 Road Map to the Proof of Theorem 15.1.1 485
15.3 The Effects of Diffusion 487
15.4 Decomposition into Viscous Traveling Waves 490
15.5 Transversal Wave Interactions 494
15.6 Interaction of Waves of the Same Family 498
15.7 Energy Estimates 503
15.8 Stability Estimates 506
15.9 Notes 508
XVI Compensated Compactness 511
16.1 The Young Measure 512
16.2 Compensated Compactness and the div curl Lemma 513
16.3 Measure Valued Solutions for Systems of Conservation Laws
and Compensated Compactness 515
16.4 Scalar Conservation Laws 518
16.5 A Relaxation Scheme for Scalar Conservation Laws 519
16.6 Genuinely Nonlinear Systems of Two Conservation Laws 523
16.7 The System of Isentropic Elasticity 525
16.8 The System of Isentropic Gas Dynamics 530
16.9 Notes 533
Bibliography 537
Author Index 613
Subject Index 621 |
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| author | Dafermos, Constantine M. 1941- |
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| author_facet | Dafermos, Constantine M. 1941- |
| author_role | aut |
| author_sort | Dafermos, Constantine M. 1941- |
| author_variant | c m d cm cmd |
| building | Verbundindex |
| bvnumber | BV022304990 |
| classification_rvk | SK 950 UF 2000 |
| classification_tum | MAT 000 PHY 210f PHY 013f MAT 357f |
| collection | ZDB-2-SMA |
| ctrlnum | (OCoLC)254564179 (DE-599)BVBBV022304990 |
| dewey-full | 530.1553535 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.1553535 |
| dewey-search | 530.1553535 |
| dewey-sort | 3530.1553535 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| discipline_str_mv | Physik Mathematik |
| doi_str_mv | 10.1007/3-540-29089-3 |
| edition | 2. ed. |
| format | Electronic eBook |
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| genre | 1\p (DE-588)4179998-7 Monografische Reihe gnd-content |
| genre_facet | Monografische Reihe |
| id | DE-604.BV022304990 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:56:26Z |
| indexdate | 2024-07-20T06:38:46Z |
| institution | BVB |
| isbn | 3540254528 9783540254522 9783540290896 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015514900 |
| oclc_num | 254564179 |
| open_access_boolean | |
| owner | DE-739 DE-355 DE-BY-UBR DE-634 DE-91 DE-BY-TUM DE-384 DE-703 DE-83 |
| owner_facet | DE-739 DE-355 DE-BY-UBR DE-634 DE-91 DE-BY-TUM DE-384 DE-703 DE-83 |
| physical | 1 Online-Ressource (XIX, 626 S.) Ill., graph. Darst. |
| psigel | ZDB-2-SMA |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Springer |
| record_format | marc |
| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Dafermos, Constantine M. 1941- Verfasser (DE-588)121360229 aut Hyperbolic conservation laws in continuum physics Constantine M. Dafermos 2. ed. Berlin [u.a.] Springer 2005 1 Online-Ressource (XIX, 626 S.) Ill., graph. Darst. txt rdacontent c rdamedia cr rdacarrier Grundlehren der mathematischen Wissenschaften 325 Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd rswk-swf Erhaltungssatz (DE-588)4131214-4 gnd rswk-swf Hyperbolisches System (DE-588)4191897-6 gnd rswk-swf Kontinuumsphysik (DE-588)4165166-2 gnd rswk-swf Erhaltungssatz (DE-588)4131214-4 s Hyperbolische Differentialgleichung (DE-588)4131213-2 s Kontinuumsphysik (DE-588)4165166-2 s DE-604 Hyperbolisches System (DE-588)4191897-6 s 1\p DE-604 Grundlehren der mathematischen Wissenschaften 325 (DE-604)BV049758308 325 https://doi.org/10.1007/3-540-29089-3 Verlag Volltext http://deposit.dnb.de/cgi-bin/dokserv?id=2669091&prov=M&dok_var=1&dok_ext=htm Beschreibung für den Leser HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015514900&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Dafermos, Constantine M. 1941- Hyperbolic conservation laws in continuum physics Grundlehren der mathematischen Wissenschaften Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd Erhaltungssatz (DE-588)4131214-4 gnd Hyperbolisches System (DE-588)4191897-6 gnd Kontinuumsphysik (DE-588)4165166-2 gnd |
| subject_GND | (DE-588)4131213-2 (DE-588)4131214-4 (DE-588)4191897-6 (DE-588)4165166-2 |
| title | Hyperbolic conservation laws in continuum physics |
| title_auth | Hyperbolic conservation laws in continuum physics |
| title_exact_search | Hyperbolic conservation laws in continuum physics |
| title_exact_search_txtP | Hyperbolic conservation laws in continuum physics |
| title_full | Hyperbolic conservation laws in continuum physics Constantine M. Dafermos |
| title_fullStr | Hyperbolic conservation laws in continuum physics Constantine M. Dafermos |
| title_full_unstemmed | Hyperbolic conservation laws in continuum physics Constantine M. Dafermos |
| title_short | Hyperbolic conservation laws in continuum physics |
| title_sort | hyperbolic conservation laws in continuum physics |
| topic | Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd Erhaltungssatz (DE-588)4131214-4 gnd Hyperbolisches System (DE-588)4191897-6 gnd Kontinuumsphysik (DE-588)4165166-2 gnd |
| topic_facet | Hyperbolische Differentialgleichung Erhaltungssatz Hyperbolisches System Kontinuumsphysik |
| url | https://doi.org/10.1007/3-540-29089-3 http://deposit.dnb.de/cgi-bin/dokserv?id=2669091&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015514900&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV049758308 |
| work_keys_str_mv | AT dafermosconstantinem hyperbolicconservationlawsincontinuumphysics |