Alinhac Hyperbolic Partial Differential Equations

1. Auflage 2009
ISBN: 978-0-387-87823-2
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 150 Seiten

Reihe: Universitext

This excellent introduction to hyperbolic differential equations is devoted to linear equations and symmetric systems, as well as conservation laws. The book is divided into two parts. The first, which is intuitive and easy to visualize, includes all aspects of the theory involving vector fields and integral curves; the second describes the wave equation and its perturbations for two- or three-space dimensions. Over 100 exercises are included, as well as 'do it yourself' instructions for the proofs of many theorems. Only an understanding of differential calculus is required. Notes at the end of the self-contained chapters, as well as references at the end of the book, enable ease-of-use for both the student and the independent researcher.

Serge Alinhac (1948-) received his PhD from l'Université Paris-Sud XI (Orsay). After teaching at l'Université Paris Diderot VII and Purdue University, he has been a professor of mathematics at l'Université Paris-Sud XI (Orsay) since 1978. He is the author of Blowup for Nonlinear Hyperbolic Equations (Birkhäuser, 1995) and Pseudo-differential Operators and the Nash-Moser Theorem (with P. Gérard, American Mathematical Society, 2007). His primary areas of research are linear and nonlinear partial differential equations.

Autoren/Hrsg.

Alinhac, Serge

Inhaltsverzeichnis

1;Contents;5
2;Introduction;9
3;Vector Fields and Integral Curves;12
3.1;1.1 First Definitions;12
3.2;1.2 Flows;13
3.3;1.3 Directional Derivatives;14
3.4;1.4 Level Surfaces;15
3.5;1.5 Bracket of Two Fields;16
3.6;1.6 Cauchy Problem and Method of Charac-teristics;16
3.7;1.7 Stopping Time;19
3.8;1.8 Straightening Out of a Field;19
3.9;1.9 Propagation of Regularity;20
3.10;1.10 Exercises;20
4;Operators and Systems in the Plane;24
4.1;2.1 Operators in the Plane: First Definitions;24
4.2;2.2 Systems in the Plane: First Definitions;26
4.3;2.3 Reducing an Operator to a System;27
4.4;2.4 Gronwall Lemma;29
4.5;2.5 Domains of Determination I (A priori Es-timate);30
4.6;2.6 Domains of Determination II (Existence);32
4.7;2.7 Exercises;33
5;Nonlinear First Order Equations;37
5.1;3.1 Quasilinear Scalar Equations;37
5.2;3.2 Eikonal Equations;40
5.3;3.3 Exercises;45
5.4;3.4 Notes;50
6;Conservation Laws in One- Space Dimension;51
6.1;4.1 First Definitions and Examples;51
6.2;4.2 Examples of Singular Solutions;52
6.3;4.3 Simple Waves;54
6.4;4.4 Rarefaction Waves;56
6.5;4.5 Riemann Invariants;57
6.6;4.6 Shock Curves;58
6.7;4.7 Lax Conditions and Admissible Shocks;60
6.8;4.8 Contact Discontinuities;62
6.9;4.9 Riemann Problem;63
6.10;4.10 Viscosity and Entropy;64
6.11;4.11 Exercises;68
6.12;4.12 Notes;72
7;The Wave Equation;74
7.1;5.1 Explicit Solutions;74
7.2;5.2 Geometry of The Wave Equation;79
7.3;5.3 Exercises;88
7.4;5.4 Notes;91
8;Energy Inequalities for the Wave Equation;92
8.1;6.1 Standard Inequality in a Strip;92
8.2;6.2 Improved Standard Inequality;95
8.3;6.3 Inequalities in a Domain;97
8.4;6.4 General Multipliers;99
8.5;6.5 Morawetz Inequality;101
8.6;6.6 KSS Inequality;103
8.7;6.7 Conformal Inequality;107
8.8;6.8 Exercises;111
8.9;6.9 Notes;119
9;Variable Coefficient Wave Equations and Systems;120
9.1;7.1 What is a Wave Equation?;120
9.2;7.2 Energy Inequality for the Wave Equation;121
9.3;7.3 Symmetric Systems;124
9.4;7.4 Finite Speed of Propagation;127
9.5;7.5 Klainerman’s Method;129
9.6;7.6 Existence of Smooth Solutions;132
9.7;7.7 Geometrical Optics;135
9.8;7.8 Exercises;139
9.9;7.9 Notes;144
10;Appendix;146
10.1;A.1 Ordinary Differential Equations;146
10.2;A.2 Submanifolds;150
11;References;156
12;Index;158