Notes on Boussinesq Equation

32 3. NONLINEAR PROBLEM. LOCAL THEORY
Moreover, for any T 0 < T there exists a neighborhood W of (f, ∂ x h) ∈ H 1 (R) × L 2 (R),
where the map
(f, h ′ ) ↦→ u
is Lipschitz from W to
{
u ∈ C([0, T0 ] : H 1 ) ∩ L 4 ([0, T 0 ] : L ∞ 1 ) / (−∆) −1/2 ∂ t u ∈ C([0, T 0 ] : L 2 ) } .
Proof. Following the ideas in the proof of the L 2 ×Ḣ−1 case, we will use a contraction
mapping argument and the estimates in section 2 to prove the first part of this theorem.
The proposition 3.6 assures that Φ(u)(t) : YT a → Y T a , so we only need to show that Φ is a
contraction.
Let u and v in YT a, with data f and g = ∂ xh, by the definition of Φ it follows that
∫t
( )
Φ(u) − Φ(v) (t) = −
Using a similar argument as in (3.8), and (3.13) we obtain
‖Φ(u) − Φ(v)‖ L 4
T L ∞ x
≤ CT sup
[0,T ]
0
V 2 (t − τ) ( |u| α u − |v| α v ) (τ) dτ. (3.21)
xx
(
‖(u − v)(t)‖ 1,2 sup ‖u‖ α 1,2 + sup ‖v‖ α )
1,2
[0,T ]
[0,T ]
To estimate ∂ x Φ(u) − ∂ x Φ(v) in L 4 T L∞ x -norm we will use the following inequality
(3.22)
||u| α u x − |v| α v x | ≤ c {( |u| α−1 + |v| α−1) |u − v||u x | + |v| α |u x − v x | } . (3.23)
Then combining the estimate (2.38), the inequality (3.23), Sobolev embedding and
Young’s inequality we obtain
‖∂ x Φ(u) − ∂ x Φ(v)‖ L 4
T L ∞ x
∫ T
≤ c
0
≤ c T ( sup
[0,T ]
∫T
≤ c
0
‖|u| α u x − |v| α v x ‖ 2 dτ
‖|u| α−1 + |v| α−1 ‖ ∞ ‖u − v‖ ∞ ‖u x ‖ 2 dτ + c
‖u(t)‖ α 1,2 + sup
[0,T ]
‖v(t)‖ α 1,2)
sup
[0,T ]
Combining (3.22) and (3.24) we have
||Φ(u) − Φ(v)||| 1
≤ c T sup
[0,T ]
‖(u − v)(t)‖ 1,2
{
2 sup
[0,T ]
≤ c T (4a α ) sup ‖(u − v)(t)‖ 1,2 .
[0,T ]
∫ T
0
‖(u − v)(t)‖ 1,2 .
‖|v| α ‖ ∞ ‖u x − v x ‖ 2 dτ
‖u(t)‖ α 1,2 + 2 sup ‖v(t)‖ α }
1,2
[0,T ]
Following the arguments in (3.13) and (3.18) it is obtained
sup
[0,T ]
‖(u − v)(t)‖ 2 ≤ c T sup
[0,T ]
‖(u − v)(t)‖ 1,2
{
sup
[0,T ]
‖u(t)‖ α 1,2 + sup ‖v(t)‖ α 1,2
[0,T ]
(3.24)
(3.25)
}
.
(3.26)
The estimate (2.37), inequality (3.23), Sobolev embedding and Young’s inequality give