Hyperbolic conservation laws in continuum physics

著者

• Dafermos, C. M. (Constantine M.)

書誌事項

Hyperbolic conservation laws in continuum physics

Constantine M. Dafermos

（Die Grundlehren der mathematischen Wissenschaften, v. 325）

Springer, c2016

4th ed

注記

Includes bibliographical references (p. 691-810) and indexes

内容説明・目次

内容説明

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

目次

• I Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Formulation of the Balance Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Reduction to Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Change of Coordinates and a Trace Theorem . . . . . . . . . . . . . . . . 71.4 Systems of Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 Companion Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6 Weak and Shock Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.7 Survey of the Theory of BV Functions . . . . . . . . . . . . . . . . . . . . . . 171.8 BV Solutions of Systems of Balance Laws . . . . . . . . . . . . . . . . . . 211.9 Rapid Oscillations and the Stabilizing Effect of CompanionBalance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23II Introduction to Continuum Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Balance Laws in Continuum Physics . . . . . . . . . . . . . . . . . . . . . . . 282.3 The Balance Laws of Continuum Thermomechanics . . . . . . . . . . 312.4 Material Frame Indifference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.5 Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.6 Thermoviscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.7 Incompressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.8 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49III Hyperbolic Systems of Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.1 Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Entropy-Entropy Flux Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3 Examples of Hyperbolic Systems of Balance Laws . . . . . . . . . . . 563.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73IV The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.1 The Cauchy Problem: Classical Solutions . . . . . . . . . . . . . . . . . . . 774.2 Breakdown of Classical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 804.3 The Cauchy Problem: Weak Solutions . . . . . . . . . . . . . . . . . . . . . . 824.4 Nonuniqueness of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 834.5 Entropy Admissibility Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 844.6 The Vanishing Viscosity Approach . . . . . . . . . . . . . . . . . . . . . . . . . 904.7 Initial-Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 944.8 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107V Entropy and the Stability of Classical Solutions . . . . . . . . . . . . . . . . . 1115.1 Convex Entropy and the Existence of Classical Solutions . . . . . . 1125.2 Relative Entropy and the Stability of Classical Solutions . . . . . . 1225.3 Involutions and Contingent Entropies . . . . . . . . . . . . . . . . . . . . . . . 1255.4 Contingent Entropies and Polyconvexity . . . . . . . . . . . . . . . . . . . . 1385.5 The Role of Damping and Relaxation . . . . . . . . . . . . . . . . . . . . . . 1465.6 Initial-Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 1605.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170VI The L1 Theory for Scalar Conservation Laws . . . . . . . . . . . . . . . . . . . 1756.1 The Cauchy Problem: Perseverance and Demiseof Classical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1766.2 Admissible Weak Solutions and their Stability Properties . . . . . . 1786.3 The Method of Vanishing Viscosity . . . . . . . . . . . . . . . . . . . . . . . . 1836.4 Solutions as Trajectories of a Contraction Semigroup and theLarge Time Behavior of Periodic Solutions . . . . . . . . . . . . . . . . . . 1886.5 The Layering Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1956.6 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1996.7 A Kinetic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2056.8 Fine Structure of LY Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2126.9 Initial-Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 2156.10 The L1 Theory for Systems of Conservation Laws . . . . . . . . . . . . 2206.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223VII Hyperbolic Systems of Balance Laws in One-Space Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2277.1 Balance Laws in One-Space Dimension . . . . . . . . . . . . . . . . . . . . 2277.2 Hyperbolicity and Strict Hyperbolicity . . . . . . . . . . . . . . . . . . . . . 2357.3 Riemann Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2387.4 Entropy-Entropy Flux Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2437.5 Genuine Nonlinearity and Linear Degeneracy . . . . . . . . . . . . . . . . 2457.6 Simple Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2477.7 Explosion of Weak Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2527.8 Existence and Breakdown of Classical Solutions . . . . . . . . . . . . . 2537.9 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2577.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 VIII Admissible Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2638.1 Strong Shocks, Weak Shocks, and Shocks of Moderate Strength 2638.2 The Hugoniot Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266<8.3 The Lax Shock Admissibility Criterion
• Compressive, Overcompressive and Undercompressive Shocks . 2728.4 The Liu Shock Admissibility Criterion . . . . . . . . . . . . . . . . . . . . . 2788.5 The Entropy Shock Admissibility Criterion . . . . . . . . . . . . . . . . . . 2808.6 Viscous Shock Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2858.7 Nonconservative Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2968.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297IX Admissible Wave Fans and the Riemann Problem. . . . . . . . . . . . . . . . 3039.1 Self-Similar Solutions and the Riemann Problem . . . . . . . . . . . . . 3039.2 Wave Fan Admissibility Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 3079.3 Solution of the Riemann Problem via Wave Curves . . . . . . . . . . . 3099.4 Systems with Genuinely Nonlinearor Linearly Degenerate Characteristic Families . . . . . . . . . . . . . . . 3129.5 General Strictly Hyperbolic Systems . . . . . . . . . . . . . . . . . . . . . . . 3169.6 Failure of Existence or Uniqueness
• Delta Shocks and Transitional Waves . . . . . . . . . . . . . . . . . . . . . . . 320^lt
• 9.7 The Entropy Rate Admissibility Criterion . . . . . . . . . . . . . . . . . . . 3239.8 Viscous Wave Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3329.9 Interaction of Wave Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3439.10 Breakdown of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3509.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354X Generalized Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35910.1 BV Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35910.2 Generalized Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36010.3 Extremal Backward Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 36210.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365XI Scalar Conservation Laws in One Space Dimension . . . . . . . . . . . . . 36711.1 Admissible BV Solutions and Generalized Characteristics . . . . . 36811.2 The Spreading of Rarefaction Waves . . . . . . . . . . . . . . . . . . . . . . . 37111.3 Regularity of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37211.4 Divides, Invariants and the Lax Formula . . . . . . . . . . . . . . . . . . . . 37711.5 Decay of Solutions Induced by Entropy Dissipation . . . . . . . . . . 38011.6 Spreading of Characteristics and Development of N-Waves . . . . 38311.7 Confinement of Characteristicsand Formation of Saw-toothed Profiles . . . . . . . . . . . . . . . . . . . . . 38411.8 Comparison Theorems and L1 Stability . . . . . . . . . . . . . . . . . . . . . 38611.9 Genuinely Nonlinear Scalar Balance Laws . . . . . . . . . . . . . . . . . . 39511.10 Balance Laws with Linear Excitation . . . . . . . . . . . . . . . . . . . . . . . 39911.11 An Inhomogeneous Conservation Law. . . . . . . . . . . . . . . . . . . . . . 401 11.12 When Genuine Nonlinearity Fails . . . . . . . . . . . . . . . . . . . . . . . . . . 40611.13 Entropy Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41811.14 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422XII Genuinely Nonlinear Systems of Two Conservation Laws . . . . . . . . . 42712.1 Notation and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42712.2 Entropy-Entropy Flux Pairs and the Hodograph Transformation 42912.3 Local Structure of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43212.4 Propagation of Riemann InvariantsAlong Extremal Backward Characteristics . . . . . . . . . . . . . . . . . . 43512.5 Bounds on Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45212.6 Spreading of Rarefaction Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 46412.7 Regularity of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46912.8 Initial Data in L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47112.9 Initial Data with Compact Support . . . . . . . . . . . . . . . . . . . . . . . . . 475<12.10 Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48112.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486XIII The Random Choice Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48913.1 The Construction Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48913.2 Compactness and Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49213.3 Wave Interactions in Genuinely Nonlinear Systems . . . . . . . . . . . 49813.4 The Glimm Functional for Genuinely Nonlinear Systems . . . . . . 50013.5 Bounds on the Total Variationfor Genuinely Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 50513.6 Bounds on the Supremum for Genuinely Nonlinear Systems . . . 50713.7 General Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50913.8 Wave Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51213.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515XIV The Front Tracking Method and Standard Riemann Semigroups . . 51714.1 Front Tracking for Scalar Conservation Laws . . . . . . . . . . . . . . . . 51814.2 Front Tracking for Genuinely NonlinearSystems of Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52014.3 The Global Wave Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52514.4 Approximate Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52614.5 Bounds on the Total Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52814.6 Bounds on the Combined Strength of Pseudoshocks . . . . . . . . . . 53114.7 Compactness and Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53414.8 Continuous Dependence on Initial Data . . . . . . . . . . . . . . . . . . . . . 53614.9 The Standard Riemann Semigroup . . . . . . . . . . . . . . . . . . . . . . . . . 54014.10 Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54114.11 Continuous Glimm Functionals,Spreading of Rarefaction Waves,and Structure of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 14.12 Stability of Strong Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55014.13 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552XV Construction of BV Solutions by the Vanishing Viscosity Method . . 55715.1 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55715.2 Road Map to the Proof of Theorem 15.1.1 . . . . . . . . . . . . . . . . . . . 55915.3 The Effects of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56115.4 Decomposition into Viscous Traveling Waves . . . . . . . . . . . . . . . . 56415.5 Transversal Wave Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56815.6 Interaction of Waves of the Same Family . . . . . . . . . . . . . . . . . . . . 57215.7 Energy Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57615.8 Stability Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57915.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582XVI BV Solutions for Systems of Balance Laws . . . . . . . . . . . . . . . . . . . . . . 58516.1 The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58616.2 Strong Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58916.3 Redistribution of Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59316.4 Bounds on the Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59516.5 L1 Stability Via Entropy with Conical Singularity at the Origin . 60616.6 L1 Stability when the Source is Partially Dissipative . . . . . . . . . . 60916.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622XVII Compensated Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62317.1 The Young Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62417.2 Compensated Compactness and the div-curl Lemma . . . . . . . . . . 62517.3 Measure-Valued Solutions for Systems of Conservation Lawsand Compensated Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . 62617.4 Scalar Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62917.5 A Relaxation Scheme for Scalar Conservation Laws . . . . . . . . . . 63117.6 Genuinely Nonlinear Systems of Two Conservation Laws . . . . . 63417.7 The System of Isentropic Elasticity . . . . . . . . . . . . . . . . . . . . . . . . 63717.8 The System of Isentropic Gas Dynamics . . . . . . . . . . . . . . . . . . . . 642<17.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648XVIII Steady and Self-similar Solutions in Multi-Space Dimensions . . . . . 65518.1 Self-Similar Solutions for Multidimensional ScalarConservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65518.2 Steady Planar Isentropic Gas Flow . . . . . . . . . . . . . . . . . . . . . . . . . 658<18.3 Self-Similar Planar Irrotational Isentropic Gas Flow . . . . . . . . . . 66318.4 Supersonic Isentropic Gas Flow Past a Ramp . . . . . . . . . . . . . . . . 66718.5 Regular Shock Reflection on a Wall . . . . . . . . . . . . . . . . . . . . . . . . 67218.6 Shock Collision with a Ramp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67518.7 Isometric Immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67818.8 Cavitation in Elastodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68218.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813

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詳細情報

• NII書誌ID(NCID): BB2141149X
• ISBN:
  • 9783662494493
• LCCN: 2016935976
• 出版国コード: gw
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: Berlin
• ページ数/冊数: xxxviii, 826 p.
• 大きさ: 25 cm
• 分類:
  • DC22 : 530.1553535
• 件名:
  • LCSH : Conservation laws (Physics)
  • LCSH : Differential equations, Hyperbolic
  • LCSH : Field theory (Physics)
• 親書誌ID:
  • BA00007422