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: Classical Solutions
4.2 Breakdown of Classical 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 of Damping 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 L1 Solutions
6.9 Initial-Boundary Value Problems
6.10 The Lt 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
I 1.6 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
ⅩⅢ The Random Choice Method
ⅩⅣ The Front Tracking Method and Standard Riemann Semigroups
ⅩⅤ Construction of BV Solutions by the Vanishing Viscosity Method.
ⅩⅥ Compensated Compactness
ⅩⅦ Conservation Laws in Two Space Dimensions
