1.1. The method of characteristics 11
Definition 1.1.4. For Z k 1, the set
PCk
consists of functions which
are piecewise
Ck
as the set of bounded continuous functions u on [0,T] × R
so that for α k and 1 j N, the restriction u|Wj belongs to
Ck(Wj)
and for all α k,
∂α(u|Wj
) extends to a bounded continuous function on
the closure W j. It is a Banach space with the norm
u
L∞([0,T ]×R)
+
|α|≤k
1≤j≤N+1
∂t,x
α
(
u
Wj
)
L∞(Wj )
.
The next result asserts that for piecewise smooth data with singularity
at x, the solution is piecewise smooth with its singularities restricted to the
characteristics through x.
Theorem 1.1.5. Suppose in addition to Hypothesis 1.1.2, that A has N
distinct real eigenvalues for all (t, x). If f PCk and g L∞(R) have
bounded continuous derivatives up to order k on each side of x, then the
solution u belongs to
PCk.
Sketch of Proof. The construction of u yielded an
L∞([0,T]×R)
estimate.
In addition we need estimates for the derivatives of order k on the wedge
Wj. Introduce
μk(u, σ) := u(σ)
L∞(R)
+
2≤|α|≤k 1≤j≤N+1
∂t,x
α
(
u
Wj
)
(σ)
L∞(Wj ∩{t=σ})
.
To estimate
un

un−1
use the following lemma.
Lemma 1.1.6. Assume the hypotheses of the theorem and that cj(t, x) is
one of the eigenvalues of A(t, x). Then, there is a constant C(j, T, L) so that
if f
PCk
and
(
∂t + cj(t, x) ∂x
)
w = f, w
t=0
= 0,
then w
PCk
and
μk(w, t) C μk(w, 0) +
t
0
μk(f, σ) .
Exercise 1.1.3. Prove the lemma. Then finish the proof of the theorem.
Exercise 1.1.4. Suppose that u is as in the theorem, f = 0, and that for
some 0 and j, the derivatives of u of order k are continuous across
γj {0 t }. Prove that they are continuous across γj {0 t T}.
Hints. Show that the set of times t for which the solution is
Ck
on γj
{0 t t} is both open and closed. Use finite speed.
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