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E-raamat: Hyperbolic Conservation Laws in Continuum Physics

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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

Arvustused

This book is a contribution to the fields of mathematical analysis and partial differential equations and their broad interactions with various branches of applied mathematics and continuum physics, whose value cannot be overstated. a useful resource for a variety of courses as well as an almost encyclopedic reference point for advanced researchers. It is a great achievement that will influence the PDEs and mathematical analysis community for years to come. (Marta Lewicka, Mathematical Reviews, November, 2017)

I Balance Laws
1(24)
1.1 Formulation of the Balance Law
2(1)
1.2 Reduction to Field Equations
3(4)
1.3 Change of Coordinates and a Trace Theorem
7(5)
1.4 Systems of Balance Laws
12(1)
1.5 Companion Balance Laws
13(2)
1.6 Weak and Shock Fronts
15(2)
1.7 Survey of the Theory of BV Functions
17(4)
1.8 BV Solutions of Systems of Balance Laws
21(2)
1.9 Rapid Oscillations and the Stabilizing Effect of Companion Balance Laws
23(1)
1.10 Notes
23(2)
II Introduction to Continuum Physics
25(28)
2.1 Kinematics
25(3)
2.2 Balance Laws in Continuum Physics
28(3)
2.3 The Balance Laws of Continuum Thermomechanics
31(4)
2.4 Material Frame Indifference
35(1)
2.5 Thermoelasticity
36(8)
2.6 Thermoviscoelasticity
44(3)
2.7 Incompressibility
47(1)
2.8 Relaxation
48(1)
2.9 Notes
49(4)
III Hyperbolic Systems of Balance Laws
53(24)
3.1 Hyperbolicity
53(1)
3.2 Entropy-Entropy Flux Pairs
54(2)
3.3 Examples of Hyperbolic Systems of Balance Laws
56(17)
3.4 Notes
73(4)
IV The Cauchy Problem
77(34)
4.1 The Cauchy Problem: Classical Solutions
77(3)
4.2 Breakdown of Classical Solutions
80(2)
4.3 The Cauchy Problem: Weak Solutions
82(1)
4.4 Nonuniqueness of Weak Solutions
83(1)
4.5 Entropy Admissibility Condition
84(6)
4.6 The Vanishing Viscosity Approach
90(4)
4.7 Initial-Boundary Value Problems
94(3)
4.8 Euler Equations
97(10)
4.9 Notes
107(4)
V Entropy and the Stability of Classical Solutions
111(64)
5.1 Convex Entropy and the Existence of Classical Solutions
112(10)
5.2 Relative Entropy and the Stability of Classical Solutions
122(3)
5.3 Involutions and Contingent Entropies
125(13)
5.4 Contingent Entropies and Polyconvexity
138(8)
5.5 The Role of Damping and Relaxation
146(14)
5.6 Initial-Boundary Value Problems
160(10)
5.7 Notes
170(5)
VI The L1 Theory for Scalar Conservation Laws
175(52)
6.1 The Cauchy Problem: Perseverance and Demise of Classical Solutions
176(2)
6.2 Admissible Weak Solutions and their Stability Properties
178(5)
6.3 The Method of Vanishing Viscosity
183(5)
6.4 Solutions as Trajectories of a Contraction Semigroup and the Large Time Behavior of Periodic Solutions
188(7)
6.5 The Layering Method
195(4)
6.6 Relaxation
199(6)
6.7 A Kinetic Formulation
205(7)
6.8 Fine Structure of L∞ Solutions
212(3)
6.9 Initial-Boundary Value Problems
215(5)
6.10 The L1 Theory for Systems of Conservation Laws
220(3)
6.11 Notes
223(4)
VII Hyperbolic Systems of Balance Laws in One-Space Dimension
227(36)
7.1 Balance Laws in One-Space Dimension
227(8)
7.2 Hyperbolicity and Strict Hyperbolicity
235(3)
7.3 Riemann Invariants
238(5)
7.4 Entropy-Entropy Flux Pairs
243(2)
7.5 Genuine Nonlinearity and Linear Degeneracy
245(2)
7.6 Simple Waves
247(5)
7.7 Explosion of Weak Fronts
252(1)
7.8 Existence and Breakdown of Classical Solutions
253(4)
7.9 Weak Solutions
257(1)
7.10 Notes
258(5)
VIII Admissible Shocks
263(40)
8.1 Strong Shocks, Weak Shocks, and Shocks of Moderate Strength
263(3)
8.2 The Hugoniot Locus
266(6)
8.3 The Lax Shock Admissibility Criterion; Compressive, Overcompressive and Undercompressive Shocks
272(6)
8.4 The Liu Shock Admissibility Criterion
278(2)
8.5 The Entropy Shock Admissibility Criterion
280(5)
8.6 Viscous Shock Profiles
285(11)
8.7 Nonconservative Shocks
296(1)
8.8 Notes
297(6)
IX Admissible Wave Fans and the Riemann Problem
303(56)
9.1 Self-Similar Solutions and the Riemann Problem
303(4)
9.2 Wave Fan Admissibility Criteria
307(2)
9.3 Solution of the Riemann Problem via Wave Curves
309(3)
9.4 Systems with Genuinely Nonlinear or Linearly Degenerate Characteristic Families
312(4)
9.5 General Strictly Hyperbolic Systems
316(4)
9.6 Failure of Existence or Uniqueness; Delta Shocks and Transitional Waves
320(3)
9.7 The Entropy Rate Admissibility Criterion
323(9)
9.8 Viscous Wave Fans
332(11)
9.9 Interaction of Wave Fans
343(7)
9.10 Breakdown of Weak Solutions
350(4)
9.11 Notes
354(5)
X Generalized Characteristics
359(8)
10.1 BV Solutions
359(1)
10.2 Generalized Characteristics
360(2)
10.3 Extremal Backward Characteristics
362(3)
10.4 Notes
365(2)
XI Scalar Conservation Laws in One Space Dimension
367(60)
11.1 Admissible BV Solutions and Generalized Characteristics
368(3)
11.2 The Spreading of Rarefaction Waves
371(1)
11.3 Regularity of Solutions
372(5)
11.4 Divides, Invariants and the Lax Formula
377(3)
11.5 Decay of Solutions Induced by Entropy Dissipation
380(3)
11.6 Spreading of Characteristics and Development of N-Waves
383(1)
11.7 Confinement of Characteristics and Formation of Saw-toothed Profiles
384(2)
11.8 Comparison Theorems and L1 Stability
386(9)
11.9 Genuinely Nonlinear Scalar Balance Laws
395(4)
11.10 Balance Laws with Linear Excitation
399(2)
11.11 An Inhomogeneous Conservation Law
401(5)
11.12 When Genuine Nonlinearity Fails
406(12)
11.13 Entropy Production
418(4)
11.14 Notes
422(5)
XII Genuinely Nonlinear Systems of Two Conservation Laws
427(62)
12.1 Notation and Assumptions
427(2)
12.2 Entropy-Entropy Flux Pairs and the Hodograph Transformation
429(3)
12.3 Local Structure of Solutions
432(3)
12.4 Propagation of Riemann Invariants Along Extremal Backward Characteristics
435(17)
12.5 Bounds on Solutions
452(12)
12.6 Spreading of Rarefaction Waves
464(5)
12.7 Regularity of Solutions
469(2)
12.8 Initial Data in L1
471(4)
12.9 Initial Data with Compact Support
475(6)
12.10 Periodic Solutions
481(5)
12.11 Notes
486(3)
XIII The Random Choice Method
489(28)
13.1 The Construction Scheme
489(3)
13.2 Compactness and Consistency
492(6)
13.3 Wave Interactions in Genuinely Nonlinear Systems
498(2)
13.4 The Glimm Functional for Genuinely Nonlinear Systems
500(5)
13.5 Bounds on the Total Variation for Genuinely Nonlinear Systems
505(2)
13.6 Bounds on the Supremum for Genuinely Nonlinear Systems
507(2)
13.7 General Systems
509(3)
13.8 Wave Tracing
512(3)
13.9 Notes
515(2)
XIV The Front Tracking Method and Standard Riemann Semigroups
517(40)
14.1 Front Tracking for Scalar Conservation Laws
518(2)
14.2 Front Tracking for Genuinely Nonlinear Systems of Conservation Laws
520(5)
14.3 The Global Wave Pattern
525(1)
14.4 Approximate Solutions
526(2)
14.5 Bounds on the Total Variation
528(3)
14.6 Bounds on the Combined Strength of Pseudoshocks
531(3)
14.7 Compactness and Consistency
534(2)
14.8 Continuous Dependence on Initial Data
536(4)
14.9 The Standard Riemann Semigroup
540(1)
14.10 Uniqueness of Solutions
541(6)
14.11 Continuous Glimm Functionals, Spreading of Rarefaction Waves, and Structure of Solutions
547(3)
14.12 Stability of Strong Waves
550(2)
14.13 Notes
552(5)
XV Construction of BV Solutions by the Vanishing Viscosity Method
557(28)
15.1 The Main Result
557(2)
15.2 Road Map to the Proof of Theorem 15.1.1
559(2)
15.3 The Effects of Diffusion
561(3)
15.4 Decomposition into Viscous Traveling Waves
564(4)
15.5 Transversal Wave Interactions
568(4)
15.6 Interaction of Waves of the Same Family
572(4)
15.7 Energy Estimates
576(3)
15.8 Stability Estimates
579(3)
15.9 Notes
582(3)
XVI BV Solutions for Systems of Balance Laws
585(38)
16.1 The Cauchy Problem
586(3)
16.2 Strong Dissipation
589(4)
16.3 Redistribution of Damping
593(2)
16.4 Bounds on the Variation
595(11)
16.5 L1 Stability Via Entropy with Conical Singularity at the Origin
606(3)
16.6 L1 Stability when the Source is Partially Dissipative
609(13)
16.7 Notes
622(1)
XVII Compensated Compactness
623(32)
17.1 The Young Measure
624(1)
17.2 Compensated Compactness and the div-curl Lemma
625(1)
17.3 Measure-Valued Solutions for Systems of Conservation Laws and Compensated Compactness
626(3)
17.4 Scalar Conservation Laws
629(2)
17.5 A Relaxation Scheme for Scalar Conservation Laws
631(3)
17.6 Genuinely Nonlinear Systems of Two Conservation Laws
634(3)
17.7 The System of Isentropic Elasticity
637(5)
17.8 The System of Isentropic Gas Dynamics
642(6)
17.9 Notes
648(7)
XVIII Steady and Self-similar Solutions in Multi-Space Dimensions
655(36)
18.1 Self-Similar Solutions for Multidimensional Scalar Conservation Laws
655(3)
18.2 Steady Planar Isentropic Gas Flow
658(5)
18.3 Self-Similar Planar Irrotational Isentropic Gas Flow
663(4)
18.4 Supersonic Isentropic Gas Flow Past a Ramp
667(5)
18.5 Regular Shock Reflection on a Wall
672(3)
18.6 Shock Collision with a Ramp
675(3)
18.7 Isometric Immersions
678(4)
18.8 Cavitation in Elastodynamics
682(4)
18.9 Notes
686(5)
Bibliography 691(120)
Author Index 811(10)
Subject Index 821
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). 
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