Tai-Ping Liu

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.

Here is an in-depth, up-to-date analysis of wave interactions for general systems of hyperbolic and viscous conservation laws. This self-contained study of shock waves explains the new wave phenomena from both a physical and a mathematical standpoint. The analysis is useful for the study of various physical situations, including nonlinear elasticity, magnetohydrodynamics, multiphase flows, combustion, and classical gas dynamic shocks. The central issue throughout the book is the understanding of nonlinear wave interactions. The book describes the qualitative theory of shock waves. It begins with the basics of the theory for scalar conservation laws and Lax's solution of the Reimann problem. For hyperbolic conservation law, the Glimm scheme and wave tracing techniques are presented and used to study the regularity and large-time behavior of solutions. Viscous nonlinear waves are studied via the recent approach to point wise estimates.

These lecture notes of the courses presented at the first CIME session 1994 by leading scientists present the state of the art in recent mathematical methods in Nonlinear Wave Propagation.