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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