26 1. Simple Examples of Propagation

Here, hj(t, ζ) is a polynomial in t, ζ. Injecting in the formula for Aexact(, t, x)

yields an expansion

Aexact(, t, x) ∼ A0(t, x) + A1(t, x) +

2A2(t,

x) + · · · ,

A0(t, x) = γ(x − e1t)/2,

(1.4.11)

Aj =

1

(2π)−d/2 2

ˆ(ζ) γ

ei(xζ−tζ1)

hj(t, ζ) dζ

=

1

2

hj(t, ∂/i)γ (x − e1t) .

(1.4.12)

The series is asymptotic as → 0 in the sense of Taylor series. For any s, N,

truncating the series after N terms yields an approximate amplitude which

differs from Aexact by O(

N+1)

in

Hs

uniformly on compact time intervals.

Exercise 1.4.2. Compute the precise form of the first corrector a1.

Formula (1.4.11) implies that if the Cauchy data are supported in a set

O, then the amplitudes Aj are all supported in the tube of rays

(1.4.13) T := (t, x) : x = x + te1, x ∈ O .

Warning 1. Though the Aj are supported in this tube, it is not true, when

d ≥ 2, that Aexact is supported in the tube. If it were, then u would be

supported in the tube. When d ≥ 2, the function u = 0 is the only solution

of D’Alembert’s equation with support in a tube of rays with compact cross

section (see Exercise 5.2.9).

Warning 2. By a closer inspection of (1.4.11) or by the analysis after

Exercise 5.2.9, one can show that Aj(t, ·)

L∞

∼

(j!)−1

∑

|β|≤2j

∂βγ||L∞

.

So, for a typical analytic γ, the series

∑

jAj

have terms of size

jCj(2j)!/j!

so they diverge no matter how small is . For a nonanalytic γ, for example

γ ∈ C0

∞,

matters are worse still. The series

∑

jAj

is a divergent series

that gives an accurate asymptotic expansion as → 0. It is a nonconvergent

Taylor expansion of Aexact(, t, x).

To analyze the oscillatory initial value problem with u(0) = 0, ut(0) =

β(x)

eix1/

requires one more idea to handle the contributions from ξ ≈ 0 in

the expression

u(t, x) =

(2π)−d/2

sin t|ξ|

|ξ|

ˆ

β ξ −

e1

eixξ

dξ .

Choose χ ∈ C0

∞(Rξ d)

with χ = 1 on a neighborhood of ξ = 0. The cutoff

integrand is equal to

χ(ξ)

sin t|ξ|

|ξ|

1

ξ − e1/

s

ks(ξ −e1/)

eixξ

, ks(ξ) := ξ

s

ˆ(ξ)

β ∈

L2(Rξ d)

.