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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 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.
ReadershipGraduate students and researchers interested in hyperbolic PDE with applications to fluid dynamics.
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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
-
Well-posedness theory
-
Viscosity
-
Relaxation
-
Nonlinear resonance
-
Multi-dimensional gas flows
-
Concluding remarks
-
-
Additional Material
-
Reviews
-
This book is recommended primarily to researchers and doctoral students. It is a unique reference about the well-posedness 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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RequestsReview Copy – for publishers of book reviewsDesk Copy – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a coursePermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
- Book Details
- Table of Contents
- Additional Material
- Reviews
- Requests
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.
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
-
Well-posedness theory
-
Viscosity
-
Relaxation
-
Nonlinear resonance
-
Multi-dimensional gas flows
-
Concluding remarks
-
This book is recommended primarily to researchers and doctoral students. It is a unique reference about the well-posedness 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
