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) dσ .

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.