xii Preface
The auditors included many at the beginnings of their careers, and I
would like to thank in particular R. Carles, E. Dumas, J. Bronski, J. Col-
liander, M. Keel, L. Miller, K. McLaughlin, R. McLaughlin, H. Zag, G.
Crippa, A. Figalli, and N. Visciglia for many interesting questions and com-
The book is aimed at the level of graduate students who have studied one
hard course in partial differential equations. Following the lead of the book
of Guillemin and Pollack (1974), there are exercises scattered throughout
the text. The reader is encouraged to read with paper and pencil in hand,
filling in and verifying as they go. There is a big difference between passive
reading and active acquisition. In a classroom setting, correcting students’
exercises offers the opportunity to teach the writing of mathematics.
To shorten the treatment and to avoid repetition with a solid partial
differential equations course, basic material such as the fundamental solution
of the wave equation in low dimensions is not presented. Naturally, I like
the treatment of that material in my book Partial Differential Equations
[Rauch, 1991].
The choice of subject matter is guided by several principles. By restrict-
ing to symmetric hyperbolic systems, the basic energy estimates come from
integration by parts. The majority of examples from applications fall under
this umbrella.
The treatment of constant coefficient problems does not follow the usual
path of describing classes of operators for which the Cauchy problem is
weakly well posed. Such results are described in Appendix 2.I along with the
Kreiss matrix theorem. Rather, the Fourier transform is used to analyse the
dispersive properties of constant coefficient symmetric hyperbolic equations
including Brenner’s theorem and Strichartz estimates.
Pseudodifferential operators are neither presented nor used. This is not
because they are in any sense vile, but to get to the core without too many
pauses to develop machinery. There are several good sources on pseudo-
differential operators and the reader is encouraged to consult them to get
alternate viewpoints on some of the material. In a sense, the expansions
of geometric optics are a natural replacement for that machinery. Lax’s
parametrix and ormander’s microlocal propagation of singularities theo-
rem require the analysis of oscillatory integrals as in the theory of Fourier
integral operators. The results require only the method of nonstationary
phase and are included.
The topic of caustics and caustic crossing is not treated. The sharp
linear results use more microlocal machinery and the nonlinear analogues
are topics of current research. The same is true for supercritical nonlinear
geometric optics which is not discussed. The subjects of dispersive and
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