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Book DetailsGraduate Studies in MathematicsVolume: 215; 2021; 437 ppMSC: Primary 35; 76;
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 multivariable 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 wellposedness 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.ReadershipGraduate students and researchers interested in hyperbolic PDE with applications to fluid dynamics.

Table of Contents

Chapters

Introduction

Preliminaries

Scalar convex conservation laws

Burgers equation

General scalar conservation laws

System of hyperbolic conservation laws, general theory

Riemann problem

Wave interactions

Wellposedness theory

Viscosity

Relaxation

Nonlinear resonance

Multidimensional gas flows

Concluding remarks


Additional Material

Reviews

This book is recommended primarily to researchers and doctoral students. It is a unique reference about the wellposedness theory of the Cauchy problem for hyperbolic systems of conservation laws. It gives a unified presentation of the program carried out by Liu and his collaborators (often former students of him), which was so far disseminated into dozens [of] papers, if not hundreds.
Denis Serre, École Normale Supérieure de Lyon


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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 multivariable 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 wellposedness 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.
Graduate students and researchers interested in hyperbolic PDE with applications to fluid dynamics.

Chapters

Introduction

Preliminaries

Scalar convex conservation laws

Burgers equation

General scalar conservation laws

System of hyperbolic conservation laws, general theory

Riemann problem

Wave interactions

Wellposedness theory

Viscosity

Relaxation

Nonlinear resonance

Multidimensional gas flows

Concluding remarks

This book is recommended primarily to researchers and doctoral students. It is a unique reference about the wellposedness theory of the Cauchy problem for hyperbolic systems of conservation laws. It gives a unified presentation of the program carried out by Liu and his collaborators (often former students of him), which was so far disseminated into dozens [of] papers, if not hundreds.
Denis Serre, École Normale Supérieure de Lyon