P.2. A bird’s eye view of hyperbolic equations xv Well posed Cauchy problems with finite speed lead to hyperbolic equa- tions.2 Since the fundamental laws of physics must respect the principles of relativity, finite speed is required. This together with causality requires hyperbolicity. Thus there are many equations from physics. Those which are most fundamental tend to have close relationships with Lorentzian ge- ometry. D’Alembert’s wave equation and the Maxwell equations are two examples. Problems with origins in general relativity are of increasing in- terest in the mathematical community, and it is the hope of hyperbolicians that the wealth of geometric applications of elliptic equations in Riemann- ian geometry will one day be paralleled by Lorentzian cousins of hyperbolic type. A source of countless mathematical and technological problems of hyper- bolic type are the equations of inviscid compressible fluid dynamics. Lin- earization of those equations yields linear acoustics. It is common that viscous forces are important only near boundaries, and therefore for many phenomena inviscid theories suﬃce. Inviscid models are often easier to com- pute numerically. This is easily understood as a small viscous term 2 ∂2/∂x2 introduces a length scale ∼ , and accurate numerics require a discretization small enough to resolve this scale, say ∼/10. In dimensions 1+d discretiza- tion of a unit volume for times of order 1 on such a scale requires 104 −4 mesh points. For only modestly small, this drives computations beyond the practical. Faced with this, one can employ meshes which are only lo- cally fine or try to construct numerical schemes which resolve features on longer scales without resolving the short scale structures. Alternatively, one can use asymptotic methods like those in this book to describe the bound- ary layers where the viscosity cannot be neglected (see for example [Grenier and Gu` es, 1998] or [G´ erard-Varet, 2003]). All of these are active areas of research. One of the key features of inviscid fluid dynamics is that smooth large solutions often break down in finite time. The continuation of such solutions as nonsmooth solutions containing shock waves satisfying suitable conditions (often called entropy conditions) is an important subarea of hyperbolic the- ory which is not treated in this book. The interested reader is referred to the conservation law references cited earlier. An interesting counterpoint is that for suitably dispersive equations in high dimensions, small smooth data yield global smooth (hence shock free) solutions (see §6.7). The subject of geometric optics is a major theme of this book. The subject begins with the earliest understanding of the propagation of light. Observation of sunbeams streaming through a partial break in clouds or a 2See [Lax, 2006] for a proof in the constant coeﬃcient linear case. The necessity of hyper- bolicity in the variable coeﬃcient case dates to [Lax, Duke J., 1957] for real analytic coeﬃcients. The smooth coeﬃcient case is due to Mizohata and is discussed in his book.

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