Notes on Boussinesq Equation

36 3. NONLINEAR PROBLEM. LOCAL THEORY
Proof. From the definition of Φ and estimate (2.38)
‖Φ(u)‖ L 4
T L ∞ ≤ ‖V 1(t)f‖
x L 4
T L ∞ + ‖V 2(t)h ′ ‖
x L 4
T L ∞ x
+ c(sup ‖(u − u 0 )(t)‖ 2 + sup ‖u 0 (t)‖ 2 )‖u‖ 4 L 4 T
[0,T ]
[0,T ] L∞ x
≤ 2δ + c sup‖(u − u 0 )(t)‖ 2 ‖u‖ 4 L 4 T
[0,T ]
L∞ x
≤ 2δ + ca 4 (a + M)
where M = sup ‖u 0 (t)‖ 2 .
[0,T ]
Choosing a = 4δ and then δ such that
we have that
+ c sup
[0,T ]
‖u 0 (t)‖ 2 ‖u‖ 4 L 4 T L∞ x
(3.34)
2 7 δ 3 c(4δ + M) < 1 (3.35)
‖Φ(u)‖ L 4
T L ∞ x ≤ 4δ.
On the other hand, the estimate (2.37) gives
sup ‖(Φ(u) − u 0 )(t)‖ 2 ≤ c sup ‖u(t)‖ 2 ‖u‖ 4 L 4 T
[0,T ]
[0,T ] L∞ x
≤ c sup ‖(u − u 0 )(t)‖ 2 ‖u‖ 4 L 4 T
[0,T ]
L∞ x
≤ a 4 c (a + M).
Taking a = 4δ with δ such that
it follows that
This shows that Φ(u(t)) : Z a T ↦→ Za T .
Now we will prove the theorem
+ c sup
[0,T ]
‖u 0 (t)‖ 2 ‖u‖ 4 L 4 T L∞ x
(3.36)
2 6 δ 3 c (4δ + M) < 1 (3.37)
sup ‖(Φ(u) − u 0 )(t)‖ 2 ≤ 4δ
[0,T ]
Proof of Theorem 3.9. Applying estimate (2.38), mean value theorem, we obtain
the following chain of inequalities.
‖Φ(u) − Φ(v)‖ L 4
T L ∞ x
∫T
≤
≤ c
0
∫ T
0
‖u 5 − v 5 ‖ 2 dτ
‖u 4 + v 4 ‖ ∞ ‖u − v‖ 2 dτ
(
≤ c sup ‖(u − v)(t)‖ 2 ‖u‖
4
L 4 T
[0,T ]
L∞ x
≤ 2a 4 c sup ‖(u − v)(t)‖ 2 .
[0,T ]
+ ‖v‖4 L 4 T L∞ x
)
□
(3.38)