24 1. Simple Examples of Propagation
Next, analyze the error in (1.4.6). Reintroduce the subscripts. Then
extract the rapidly oscillating factor in (1.4.5) to find exact amplitudes,
u±(t, x) =
ei(x1∓t)/
Aexact(,
±
t, x) ,
Aexact(,
±
t, x) :=
1
(2π)d/22
ˆ(ζ) γ
eix.ζ e∓it(|e1+ζ|−1)/
dζ . (1.4.7)
Proposition 1.4.2. The exact and approximate solutions of u = 0 with
Cauchy data (1.4.4) are given by
u =
±
ei(x1∓t)/
Aexact(,
±
t, x) , uapprox =
±
ei(x1∓t)/
γ(x ∓ e1t)
2
,
as in (1.4.7) and (1.4.6). The error is O( ) on bounded time intervals.
Precisely, there is a constant C 0 so that for all s, , t,
Aexact(,
±
t, x) −
γ(x ∓ e1t)
2
Hs(RN )
≤ C |t| γ
Hs+2(Rd)
.
Proof. It suffices to estimate the error with the plus sign. The definitions
yield
Aexact(,
+
t, x) − γ(x−e1t)/2 = C ˆ(ζ) γ
eix.ζ
(
e−it(|e1+ζ|−1)/)−e−itζ1
)
dζ .
The definition of the
Hs(Rd)
norm yields
(1.4.8) Aexact(,
+
t, x) − γ(x − e1t)/2
Hs(RN )
= ζ
s
ˆ(ζ) γ
(
e−it(|e1+ζ|−1)/
−
e−itζ1
)
L2(RN )
.
Taylor expansion yields for |β| ≤ 1/2,
| e1 + β | = 1 + β1 + r(β) , |r(β)| ≤ C
|β|2
.
Increasing C if needed, the same inequality is true for |β| ≥ 1/2 as well.
Applied to β = ζ this yields
t
(
e1 + ζ − 1
)
/ − t ζ1 x1 ≤ C
|t||ζ|2
so
e−it(|e1+ζ|−1)/
−
e−itζ1
≤ C
|t||ζ|2
.
Therefore,
(1.4.9)
ζ
s
ˆ(ζ) γ
(
e−it(|e1+ζ|−1)/
−
e−itζ1
)
L2(Rd)
≤ C |t| ζ
s|ζ|2
ˆ(ζ) γ
L2
.
Combining (1.4.8) and (1.4.9) yields the estimate of the proposition.
