Shock Waves (Graduate Studies in Mathematics)

This book presents the fundamentals of the shock wave theory. The first part of the book, Chapters 1 through 5, covers the basic elements of the shock wave theory by analyzing the scalar conservation laws. The main focus of the analysis is on the explicit solution behavior. This first part of the book requires only a course in multi-variable calculus, and can be used as a text for an undergraduate topics course. In the second part of the book, Chapters 6 through 9, this general theory is used to study systems of hyperbolic conservation laws. This is a most significant well-posedness theory for weak solutions of quasilinear evolutionary partial differential equations. The final part of the book, Chapters 10 through 14, returns to the original subject of the shock wave theory by focusing on specific physical models. Potentially interesting questions and research directions are also raised in these chapters. The book can serve as an introductory text for advanced undergraduate students and for graduate students in mathematics, engineering, and physical sciences. Each chapter ends with suggestions for further reading and exercises for students. Preface Chapter 1. Introduction Chapter 2. Preliminaries 1. Method of Characteristics 2. Development of Singularities 3. Weak Solutions, Rankine-Hugoniot Condition 4. Expansion Waves 5. Non-uniqueness, Entropy Condition 6. Notes 7. Exercises Chapter 3. Scalar Convex Conservation Laws 1. Riemann Problem 2. Hopf Equation 3. Wave Interactions, Constructing Solutions 4. Well-Posedness Theory 5. Generalized Characteristics, Nonlinear Regularization 6. N-Waves, Inviscid Dissipation 7. Entropy Pairs 8. Generalized Entropy Functional 9. Notes 10. Exercises Chapter 4. Burgers Equation 1. Heat Equation 2. Hopf-Cole Transformation 3. Inviscid Limit 4. Nonlinear Waves 5. Linearized Hopf-Cole Transformation 6. Green’s Functions 7. Nonlinearity 8. Metastable States 9. Notes 10. Exercises Chapter 5. General Scalar Conservation Laws 1. Viscous Shock Profiles 2. Riemann Problem 3. L1 Stability 4. Scattering Wave Patterns 5. Entropy Pairs 6. Multi-Dimensional Laws 7. Notes 8. Exercises Chapter 6. Systems of Hyperbolic Conservation Laws: General Theory 1. Hyperbolicity 2. Entropy and Symmetry 3. Symmetry and Energy Estimate 4. Local Existence of Smooth Solutions 5. Euler Equations in Gas Dynamics 6. Shock Waves 7. Notes 8. Exercises Chapter 7. Riemann Problem 1. Linear System 2. Simple Waves 3. Hugoniot Curves 4. Riemann Problem I 5. Examples I 6. Riemann Problem II 7. Examples II 8. Notes 9. Exercises Chapter 8. Wave Interactions 1. Interaction of Infinitesimal Waves 2. A 2×2 System and Coordinates of Riemann Invariants 3. A 3×3 System 4. General Analysis 5. Notes 6. Exercises Chapter 9. Well-Posedness Theory 1. Glimm Scheme 2. Nonlinear Functional 3. Wave Tracing 4. Existence Theory 5. Stability Theory 6. Generalized Characteristics and Expansion of Rarefaction Waves 7. Large-Time Behavior 8. Regularity 9. Decay and N-Waves 10. Some Basics of Numerical Computations 11. Notes 12. Exercises Chapter 10. Viscosity 1. Nonlinear Waves for Scalar Laws 2. Wave Interaction for Systems 3. Physical Models 4. The p-System 5. General Dissipative Systems 6. Notes 7. Exercises Chapter 11. Relaxation 1. A Simple Relaxation Model 2. Examples 3. Gas In Thermal Non-equilibrium 4. The Boltzmann Equation in Kinetic Theory 5. Notes 6. Exercises Chapter 12. Nonlinear Resonance 1. Moving Source 2. Sub-shocks 3. Non-strict Hyperbolicity 4. Vacuum 5. Boundary 6. Kinetic Boundary Layers and Fluid-like Waves 7. Shock Profiles for Difference Schemes 8. Notes 9. Exercises Chapter 13. Multi-Dimensional Gas Flows 1. Linear Waves 2. Discontinuity Waves 3. Potential Flows 4. Self-Similar Flows and the Ellipticity Principle 5. Characteristics and Simple Waves 6. Hodograph Transformation 7. The Shock Polar 8. Prandtl Paradox 9. Notes 10. Exercises Chapter 14. Concluding Remarks 1. Development of Singularities 2. Local and Global Behavior for Gas Flows with Shock Waves 3. Nonlinear Waves for Viscous Conservation Laws 4. Well-Posedness Theory for Weak Solutions 5. Kinetic Theory and Fluid Dynamics 6. Multiple Effects Bibliography Index "This book presents the fundamentals of the shock wave theory. Shock waves are present in many natural situations as a consequence of nonlinear constitutive relations. Mathematical analysis of shock waves is based mostly on the conservation laws. Consider the case of gas dynamics. There is the conservation of mass. The conservation of momentum follows from the Newtonian physics. During the nineteenth century, the conservation of energy and the second law of thermodynamics were formulated. This is the scientific background when Stokes [119] and Riemann [112] did their pioneering works on shock waves in the mid-nineteenth century. More conservation laws were subsequently formulated with the study of electro-magnetism, nonlinear elasticity, high temperature gas dynamics, and other physical phenomena. There have been important, continuing progresses on the development of shock wave theory since the time of Stokes and Riemann."-- Preface