Hyperbolic Partial Differential Equations

1. Front Matter

   Pages i-xi

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2. Vector Fields and Integral Curves

   • Serge Alinhac

   Pages 1-12

3. Operators and Systems in the Plane

   • Serge Alinhac

   Pages 13-25

4. Nonlinear First Order Equations

   • Serge Alinhac

   Pages 27-40

5. Conservation Laws in One-Space Dimension

   • Serge Alinhac

   Pages 41-63

6. The Wave Equation

   • Serge Alinhac

   Pages 65-82

7. Energy Inequalities for the Wave Equation

   • Serge Alinhac

   Pages 83-110

8. Variable Coefficient Wave Equations and Systems

   • Serge Alinhac

   Pages 111-136

9. Back Matter

   Pages 1-14

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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. ).

• Vector field
• conservation laws
• hyperbolic partial differential equation
• integral curves
• partial differential equation
• submanifolds
• vector fields
• wave equation
• wave equations
• partial differential equations

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 Kopáčková, 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 für Mathematik, Vol. 163 (1), May, 2011)

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.

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