12 1. Simple Examples of Propagation
Denote by Φj(t, x) the flow of the ordinary differential equation x =
cj(t, x). That is x(t) = Φj(t, x) is the solution with x(0) = x. The solution
operator for the pure transport equation (∂t + cj∂x)u = 0 with initial value
g is then
u(t) = g(Φj(−t, x)) .
The values at time t are the rearrangements by the diffeomorphism Φ(−t, ·)
of the initial function. Because of the uniform boundedness of the derivatives
of cj on slabs [0,T] × R, one has
× R) .
The derivative ∂xΦ measures the expansion or contraction by the flow. It is
the length of the image of an infinitesimal interval divided by the original
length. In particular Φ can at most expand lengths by a bounded quan-
tity. The inverse of Φ(t, ·) is the flow by the ordinary differential equation
from time t to time 0, so the inverse also cannot expand by much. This is
equivalent to a lower bound,
× R) .
The diffeomorphism Φ(t, ·) can neither increase nor decrease length by much.
Therefore the maps u(0) → u(t) are uniformly bounded maps from
itself for all p ∈ [1, ∞]. The case p = ∞ is equivalent to the Haar inequalities.
There are analogous estimates
≤ C u(t)
with constant independent of p. This in turn leads to an existence theory like
that just recounted but mk(u, t) is replaced by
these one dimensional hyperbolic Cauchy problems, there is a wide class of
spaces for which the evolution is well posed. The case of p = 1 is particularly
important for the theory of shock waves in d = 1. Brenner’s Theorem 3.3.5
shows that only the case p = 2 remains valid for typical hyperbolic equations
in dimension d 1.
1.2. Examples of propagation of singularities using
D’Alembert’s solution (see Example 1.1.2) of the one-dimensional wave equa-
(1.2.1) utt − uxx = 0 ,
is the sum of progressing waves
(1.2.2) f(x − t) + g(x + t) .