Notes on Boussinesq Equation

30 3. NONLINEAR PROBLEM. LOCAL THEORY
To prove the local existence we will make use of the estimates obtained in Chapter 2
and a contraction mapping argument.
In this section it will be used the following notation.
||u|| 1 =
( ∫ T
0
) 1/4 ( ∫ T
‖u(t)‖ 4 ∞ dt +
Consider the following complete metric space
Y a T = { u ∈C([0, T ] : H 1 (R)) ∩ L 4 ([0, T ] : L ∞ 1 (R) ) /
sup
[0,T ]
0
‖u x (t)‖ 4 ∞ dt) 1/4.
(−∆) −1/2 ∂ t u ∈ C([0, T ] : L 2 (R)),
‖u(t)‖ 1,2 ≤ a, ||u(t)|| 1 ≤ a, sup ‖(−∆) −1/2 ∂ t u(t)‖ 2 ≤ a }
[0,T ]
Proposition 3.6. For 0 < α, f ∈ H 1 (R) and g = h ′ ∈ L 2 (R) define Φ(u)(t) as in
(3.3). Then
Φ(u)(t) : Y a T → Y a T
for some T and a depending on δ and α, where δ = max(δ 1 , δ 2 ), and ‖f‖ 1,2 ≤ δ 1 , ‖h‖ 2 ≤ δ 2
.
Proof. Using (2.28), (2.34), (2.38) in conjunction with Sobolev’s embedding theorem
we obtain
‖Φ(u)‖ L 4
T L ∞ x
. ≤ c(1 + T 1/4 )‖f‖ 2 + ‖h‖ −1,2 + c ‖u‖ α ∞‖u‖ 2 dτ
≤ c (1 + T 1/4 )‖f‖ 1,2 + ‖h‖ 2 + c T sup ‖u(t)‖ α+1
1,2
0
(3.13)
[0,T ]
∫ T
The same argument as in (3.13) gives us the following
‖∂ x Φ(u)‖ L 4
T L ∞ x
≤ c (1 + T 1/4 )(‖f‖ 1,2 + ‖h‖ 2 ) + c T sup ‖u(t)‖ α+1
1,2 . (3.14)
[0,T ]
Then from (3.13) and (3.14) it follows that
||Φ(u)|| 1 ≤ 2c (1 + T 1/4 ) ( )
‖f‖ 1,2 + ‖h‖ 2 + 2c T sup ‖u(t)‖ α+1
1,2
[0,T ]
(3.15)
≤ 2c (1 + T 1/4 )(2δ) + 2cT a α+1
choosing a = 8cδ and T such that
(T 1/4 + T 2 3α+2 c α+1 δ α ) < 1 (3.16)
it follows that
||Φ(u)|| 1 ≤ 8cδ.