Notes on Boussinesq Equation

20 2. LINEAR PROBLEM
Since φ ′ (ξ) ≠ 0, there exists ψ such that φ(ψ(ξ)) = ξ, ξ ≥ 0. Then making the change
of variables η = φ(ξ) we have that
∫
I 1 =
e itη ixψ(η) dη
e
ˆ˜f(ψ(η))
φ ′ (ψ(η)) . (2.42)
where ˜f(x) = f(x), for x < 0 and equals 0 otherwise.
Taking the L 2 –norm of I 1 in t, using Plancherel’s identity and returning to the previous
variables we have that
‖I 1 ‖ 2 L 2 t
= c
∫ ∞
−∞
∫ ∞
|e ixψ(η) ˆ˜f(ψ(η))|
2
|φ ′ (ψ(η))| 2 dη
| ˆ˜f(ψ(η))| 2
= c
|φ ′ (ψ(η))| 2 dη
(2.43)
−∞
∫ | ˆ˜f(ξ)|
2
= c
|φ ′ (ξ)| dξ.
A similar argument can be used to estimate I 2 . Hence the result follows.
□
Proposition 2.11. Let V 1 (t) and V 2 (t) be defined as in the statements of Lemmas
2.4 and 2.5, then:
For f ∈ L 2
( ∫ T
sup
x
0
|D 1/2
x V 1 (t)f(x)| 2 dt) 1/2
≤ c (1 + T 1/2 )‖f‖ 2 . (2.44)
If h ′ ∈ Ḣ−1
( ∫ T
sup
x
0
|D 1/2
x V 2 (t)h ′ (x)| 2 dt) 1/2
≤ c (1 + T 1/2 )‖h‖ −1,2 . (2.45)
And for p ∈ L 2 ( ∫ T
sup
x
0
|D 1/2
x V 2 (t)p ′′ (x)| 2 dt) 1/2
≤ c (1 + T 1/2 )‖p‖ 2 . (2.46)
Proof. To prove (2.44), (2.45) and (2.46) we shall use the same argument, so we only
will sketch the proof of (2.44).