xiv Preface owe a great intellectual debt to the lecture notes, and to all that Peter Lax has taught me through the years. The book introduces a large and rich subject and I hope that readers are sufficiently attracted to probe further. P.2. A bird’s eye view of hyperbolic equations The central theme of this book is hyperbolic partial differential equations. These equations are important for a variety of reasons that we sketch here and that recur in many different guises throughout the book. The first encounter with hyperbolicity is usually in considering scalar real linear second order partial differential operators in two variables with coefficients that may depend on x, a ux 1 x1 + b ux 1 x2 + c ux 2 x2 + lower order terms . Associate the quadratic form ξ a ξ2 1 + b ξ1ξ2 +c ξ2. 2 The differential opera- tor is elliptic when the form is positive or negative definite. The differential operator is strictly hyperbolic when the form is indefinite and nondegener- ate.1 In the elliptic case one has strong local regularity theorems and solv- ability of the Dirichlet problem on small discs. In the hyperbolic cases, the initial value problem is locally well set with data given at noncharacteristic curves and there is finite speed of propagation. Singularities or oscillations in Cauchy data propagate along characteristic curves. The defining properties of hyperbolic problems include well posed Cauchy problems, finite speed of propagation, and the existence of wave like structures with infinitely varied form. To see the latter, consider in Rt,x 2 initial data on t = 0 with the form of a short wavelength wave packet, a(x) eix/, localized near a point p. The solution will launch wave packets along each of two characteristic curves. The envelopes are computed from those of the initial data, as in §5.2, and can take any form. One can send essentially arbitrary amplitude modulated signals. The infinite variety of wave forms makes hyperbolic equations the pre- ferred mode for communicating information, for example in hearing, sight, television, and radio. The model equations for the first are the linearized compressible inviscid fluid dynamics, a.k.a. acoustics. For the latter three it is Maxwell’s equations. The telecommunication examples have the property that there is propagation with very small losses over large distances. The examples of wave packets and long distances show the importance of short wavelength and large time asymptotic analyses. 1 The form is nondegenerate when its defining symmetric matrix is invertible.
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