Hyperbolic Partial Differential Equations

By: Serge Alinhac

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Paperback | 29 June 2009

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164 Pages

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The aim of this book is to present hyperbolic partial di?erential equations at an elementary level. In fact, the required mathematical background is only a third year university course on di?erential calculus for functions of several variables. No functional analysis knowledge is needed, nor any distribution theory (with the exception of shock waves mentioned below). k All solutions appearing in the text are piecewise classical C solutions. Beyond the simpli?cations it allows, there are several reasons for this choice: First, we believe that all main features of hyperbolic partial d- ferential equations (PDE) (well-posedness of the Cauchy problem, ?nite speed of propagation, domains of determination, energy inequalities, etc. ) canbedisplayedinthiscontext. Wehopethatthisbookitselfwillproveour belief. Second, allproperties, solutionformulas, andinequalitiesestablished here in the context of smooth functions can be readily extended to more general situations (solutions in Sobolev spaces or temperate distributions, etc. ) by simple standard procedures of functional analysis or distribution theory, which are "external" to the theory of hyperbolic equations: The deep mathematical content of the theorems is already to be found in the statements and proofs of this book. The last reason is this: We do hope that many readers of this book will eventually do research in the ?eld that seems to us the natural continuation of the subject: nonlinear hyp- bolic systems (compressible ?uids, general relativity theory, etc. ).

Industry Reviews

From the reviews:

"The aim of the present book is to present hyperbolic partial differential equations at an elementary level. ... the novice might well be used to a more discursive style. ... HypPDE is a very good book ... the more experienced mathematician will also find a lot of good stuff in these pages, all presented well and cogently." (Michael Berg, The Mathematical Association of America, October, 2009)

"Any specification for the perfect mathematical monograph would doubtless exhibit self-contradiction. Where one reader requires copious details and examples, another wants a breathless flow. This terse volume on hyperbolic differential equations (which describe processes, such as wave propagation, where signals travel at finite speed) serves a ... need. ... Summing Up: Highly recommended. Upper-division undergraduate through professional collections." (D. V. Feldman, Choice, Vol. 47 (8), April, 2010)

"Provides an introduction to linear hyperbolic equations symmetric hyperbolic systems and conservation laws. ... The presentation is clear and over 100 exercises are included ... which guide the reader step by step through the proofs of theorems. ... This book provides an excellent introduction to hyperbolicequations and conservation laws, and it can be recommended to anyone who wishes to study this fertile branch of partial differential equations." (Alan Jeffrey, Mathematical Reviews, Issue 2010 e)

"The book is useful to senior or graduated students as well as to researchers of other fields interested in hyperbolic partial differential equations. ... The author often uses geometrical explanations of problems instead of tedious mathematical proofs. Some illustrative pictures would be desirable." (Marie Kopackova, Zentralblatt MATH, Vol. 1178, 2010)

"This is a concise introduction to the main aspects of the theory of hyperbolic partial differential equations. ... This work is highly recommended for a quickand smooth entry into this field of great current interest." (M. Kunzinger, Monatshefte fuer Mathematik, Vol. 163 (1), May, 2011)

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Introduction	p. ix
Vector Fields and Integral Curves	p. 1
First Definitions	p. 1
Flows	p. 2
Directional Derivatives	p. 3
Level Surfaces	p. 4
Bracket of Two Fields	p. 5
Cauchy Problem and Method of Characteristics	p. 5
Stopping Time	p. 8
Straightening Out of a Field	p. 8
Propagation of Regularity	p. 9
Exercises	p. 9
Operators and Systems in the Plane	p. 13
Operators in the Plane: First Definitions	p. 13
Systems in the Plane: First Definitions	p. 15
Reducing an Operator to a System	p. 16
Gronwall Lemma	p. 18
Domains of Determination I (A priori Estimate)	p. 19
Domains of Determination II (Existence)	p. 21
Exercises	p. 22
Nonlinear First Order Equations	p. 27
Quasilinear Scalar Equations	p. 27
Eikonal Equations	p. 30
Exercises	p. 35
Notes	p. 40
Conservation Laws in One-Space Dimension	p. 41
First Definitions and Examples	p. 41
Examples of Singular Solutions	p. 42
Simple Waves	p. 44
Rarefaction Waves	p. 46
Riemann Invariants	p. 47
Shock Curves	p. 48
Lax Conditions and Admissible Shocks	p. 50
Contact Discontinuities	p. 52
Riemann Problem	p. 53
Viscosity and Entropy	p. 54
Entropy	p. 55
Limits of Viscosity Solutions	p. 55
Application to the Case of a Shock	p. 56
Exercises	p. 58
Notes	p. 62
The Wave Equation	p. 65
Explicit Solutions	p. 65
Fourier Analysis	p. 66
Solutions as Spherical Means	p. 67
Finite Speed of Propagation, Domains of Determination	p. 68
Strong Huygens Principle	p. 69
Asymptotic Behavior of Solutions	p. 69
Geometry of The Wave Equation	p. 70
Rotations and Scaling Vector Field	p. 71
Hyperbolic Rotations and Lorentz Fields	p. 74
Light Cones, Null Frames, and Good Derivatives	p. 75
Klainerman's Inequality	p. 77
Exercises	p. 79
Notes	p. 82
Energy Inequalities for the Wave Equation	p. 83
Standard Inequality in a Strip	p. 83
Improved Standard Inequality	p. 86
Inequalities in a Domain	p. 88
General Multipliers	p. 90
Morawetz Inequality	p. 92
KSS Inequality	p. 94
Conformal Inequality	p. 98
Exercises	p. 102
Notes	p. 110
Variable Coefficient Wave Equations and Systems	p. 111
What is a Wave Equation?	p. 111
Energy Inequality for the Wave Equation	p. 112
Symmetric Systems	p. 115
Definitions and Examples	p. 115
Energy Inequality	p. 117
Finite Speed of Propagation	p. 118
Klainerman's Method	p. 120
Existence of Smooth Solutions	p. 123
Geometrical Optics	p. 126
An Algebraic Computation	p. 126
Formal and Actual Geometrical Optics	p. 127
Parametrics for the Cauchy Problem	p. 128
Exercises	p. 130
Notes	p. 135
	p. 137
Ordinary Differential Equations	p. 137
Cauchy Problem	p. 137
Flows	p. 140
Lower and Upper Fences	p. 141
Submanifolds	p. 141
First Definitions	p. 141
Submanifolds Defined by Equations	p. 142
Parametrized Surfaces	p. 143
Graphs	p. 144
Weaving	p. 145
Stokes Formula	p. 145
References	p. 147
Index	p. 149
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Hyperbolic Partial Differential Equations

At a Glance

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