Ⅰ Balance Laws

1.1 Formulation of the Balance Law

1.2 Reduction to Field Equations

1.3 Change of Coordinates and a Trace Theorem

1.4 Systems of Balance Laws

1.5 Companion Balance Laws

1.6 Weak and Shock Fronts

1.7 Survey of the Theory of BV Functions

1.8 BV Solutions of Systems of Balance Laws

1.9 Rapid Oscillations and the Stabilizing Effect of Companion Balance Laws

1.10 Notes

Ⅱ Introduction to Continuum Physics

2.1 Bodies and Motions

2.2 Balance Laws in Continuum Physics

2.3 The Balance Laws of Continuum Thermomechanics

2.4 Material Frame Indifference

2.5 Thermoelasticity

2.6 Thermoviscoelasticity

2.7 Incompressibility

2.8 Relaxation

2.9 Notes

Ⅲ Hyperbolic Systems of Balance Laws

3.1 Hyperbolicity

3.2 Entropy-Entropy Flux Pairs

3.3 Examples of Hyperbolic Systems of Balance Laws

3.4 Notes

Ⅳ The Cauchy Problem

4.1 The Cauchy Problem:Ciassical Solutions

4.2 Breakdown ofClassical Solutions

4.3 The Cauchy Problem:Weak Solutions

4.4 Nonuniqueness of Weak Solutions

4.5 Entropy Admissibility Condition

4.6 The Vanishing Viscosity Approach

4.7 Initial-Boundary Value Problems

4.8 Notes

Ⅴ Entropy and the Stability of Classical Solutions

5.1 Convex Entropy and the Existence of Classical Solutions

5.2 The Role ofDamping and Relaxation

5.3 Convex Entropy and the Stability of Classical Solutions

5.4 Involutions

5.5 Contingent Entropies and Polyconvexity

5.6 Initial-Boundary Value Problems

5.7 Notes

Ⅵ The L1 Theory for Scalar Conservation Laws

6.1 The Cauchy Problem:Perseverance and Demise of Classical Solutions

6.2 Admissible Weak Solutions and their Stability Properties

6.3 The Method of Vanishing Viscosity

6.4 Solutions as Trajectories of a Contraction Semigroup

6.5 The Layering Method

6.6 Relaxation

6.7 A Kinetic Formulation

6.8 Fine Structure of L∞ Solutions

6.9 Initial-Boundary Value Problems

6.10 The L1 Theory for Systems of Conservation Laws

6.11 Notes

Ⅶ Hyperbolic Systems of Balance Laws in One-Space Dimension

7.1 Balance Laws in One-Space Dimension

7.2 Hyperbolicity and Strict Hyperbolicity

7.3 Riemann Invariants

7.4 Entropy-Entropy Flux Pairs

7.5 Genuine Nonlinearity and Linear Degeneracy

7.6 Simple Waves

7.7 Explosion of Weak Fronts

7.8 Existence and Breakdown of Classical Solutions

7.9 Weak Solutions

7.10 Notes

Ⅷ Admissible Shocks

8.1 Strong Shocks,Weak Shocks,and Shocks of Moderate Strength

8.2 The Hugoniot Locus

8.3 The Lax Shock Admissibility Criterion;Compressive,Overcompressive and Undercompressive Shocks

8.4 The Liu Shock Admissibility Criterion

8.5 The Entropy Shock Admissibility Criterion

8.6 Viscous Shock Profiles

8.7 Nonconservative Shocks

8.8 Notes

Ⅸ Admissible Wave Fans and the Riemann Problem

9.1 Self-Similar Solutions and the Riemann Problem

9.2 Wave Fan Admissibility Criteria

9.3 Solution of the Riemann Problem via Wave Curves

9.4 Systems with Genuinely Nonlinear or Linearly Degenerate Characteristic Families

9.5 General Strictly Hyperbolic Systems

9.6 Failure of Existence or Uniqueness;Delta Shocks and Transitional Waves

9.7 The Entropy Rate Admissibility Criterion

9.8 Viscous Wave Fans

9.9 Interaction of Wave Fans

9.10 Breakdown of Weak Solutions

9.11 Notes

Ⅹ Generalized Characteristics

10.1 BV Solutions

10.2 Generalized Characteristics

10.3 Extremal Backward Characteristics

10.4 Notes

Ⅺ Genuinely Nonlinear Scalar Conservation Laws

11.1 Admissible BV Solutions and Generalized Characteristics

11.2 The Spreading of Rarefaction Waves

11.3 Regularity of Solutions

11.4 Divides,Invariants and the Lax Formula

11.5 Decay of Solutions Induced by Entropy Dissipation

11.6 Spreading of Characteristics and Development of N-Waves

11.7 Confinement of Characteristics and Formation of Saw-toothed Profiles

11.8 Comparison Theorems and L1 Stability

11.9 Genuinely Nonlinear Scalar Balance Laws

11.10 Balance Laws with Linear Excitation

11.11 An Inhomogeneous Conservation Law

11.12 Notes

Ⅻ Genuinely Nonlinear Systems of Two Conservation Laws

12.1 Notation and Assumptions

12.2 Entropy-Entropy Flux Pairs and the Hodograph Transformation

12.3 Local Structure of Solutions

12.4 Propagation of Riemann Invariants Along Extremal Backward Characteristics

12.5 Bounds on Solutions

12.6 Spreading of Rarefaction Waves

12.7 Regularity of Solutions

12.8 Initial Data in L1

12.9 Initial Data with Compact Support

12.10 Periodic Solutions

12.11 Notes

ⅩⅢ The Random Choice Method

13.1 The Construction Scheme

13.2 Compactness and Consistency

13.3 Wave Interactions,Approximate Conservation Laws and Approximate Characteristics in Genuinely Nonlinear Systems

13.4 The Glimm Functional for Genuinely Nonlinear Systems

13.5 Bounds on the Total Variation for Genuinely Nonlinear Systems

13.6 Bounds on the Supremum for Genuinely Nonlinear Systems

13.7 General Systems

13.8 Wave Tracing

13.9 Inhomogeneous Systems of Balance Laws

13.10 Notes

ⅩⅣ The Front Tracking Method and Standard Riemann Semigroups

14.1 Front Tracking for Scalar Conservation Laws

14.2 Front Tracking for Genuinely Nonlinear Systems of Conservation Laws

14.3 The Global Wave Pattern

14.4 Approximate Solutions

14.5 Bounds on the Total Variation

14.6 Bounds on the Combined Strength of Pseudoshocks

14.7 Compactness and Consistency

14.8 Continuous Dependence on Initial Data

14.9 The Standard Riemann Semigroup

14.10 Uniqueness of Solutions

14.11 Continuous Glimm Functionals,Spreading of Rarefaction Waves,and Structure of Solutions

14.12 Stability of Strong Waves

14.13 Notes

ⅩⅤ Construction of BV Solutions by the Vanishing Viscosity Method

15.1 The Main Result

15.2 Road Map to the ProofofTheorem 15.1.1

15.3 The Effects of Diffusion

15.4 Decomposition into Viscous Traveling Waves

15.5 Transversal Wave Interactions

15.6 Interaction of Waves of the Same Family

15.7 Energy Estimates

15.8 Stability Estimates

15.9 Notes

ⅩⅥ Compensated Compactness

16.1 The Young Measure

16.2 Compensated Compactness and the div-curl Lemma

16.3 Measure-Valued Solutions for Systems of Conservation Laws and Compensated Compactness

16.4 Scalar Conservation Laws

16.5 A Relaxation Scheme for Scalar Conservation Laws

16.6 Genuinely Nonlinear Systems of Two Conservation Laws

16.7 The System of Isentropic Elasticity

16.8 The System of Isentropic Gas Dynamics

16.9 Notes

ⅩⅦ Conservation Laws in Two Space Dimensions

17.1 Self-Similar Solutions for Multidimensional Scalar Conservation Laws

17.2 Steady Planar Isentropic Gas Flow

17.3 Self-Similar Planar Irrotational Isentropic Gas Flow

17.4 Supersonic Isentropic Gas Flow Past a Ramp of Gentle Slope

17.5 Regular Shock Reflection on a Wall

17.6 Shock Collision with a Steep Ramp

17.7 Notes

Bibliography

Author Index

Subject Index