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
∂t,xΦ
α
∈
L∞([0,T]
× 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,
(
∂xΦ
)−1
∈
L∞([0,T]
× 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
Lp(R)
to
itself for all p ∈ [1, ∞]. The case p = ∞ is equivalent to the Haar inequalities.
There are analogous estimates
u(t)
Lp(R)
≤ C u(t)
Lp(R)
+
t
0
Lu(σ)
Lp(R)
dσ ,
with constant independent of p. This in turn leads to an existence theory like
that just recounted but mk(u, t) is replaced by
∑
|α|≤k
∂t,xu(t)
α
Lp(R). For
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
progressing waves
D’Alembert’s solution (see Example 1.1.2) of the one-dimensional wave equa-
tion,
(1.2.1) utt − uxx = 0 ,
is the sum of progressing waves
(1.2.2) f(x − t) + g(x + t) .
