Notes on Boussinesq Equation

CHAPTER 4
Global Theory. Persistence
4.1. Global Theory in H 1
In this section our aim is to prove that for small data the solution of the IVP (1.1)
in H 1 × L 2 can be extended globally (t > 0). To show this, we combine the conservation
law (4.2) (see below) and one of the results obtained by J. Bona and R. Sachs in [6]. As
we pointed out in the introduction, Kato’s theory [22] allowed them to show the local
well-posedness of the system (1.5)-(1.6) for smooth data, as we will see below.
We begin by stating the following result of Bona and Sachs in [6].
Denote by
Y s (T ) = C ( [0, T ] : H s+2 (R) ) ∩ C 1( [0, T ] : H s (R) ) ∩ C 2 ([0, T ] : H s−2 (R)).
Theorem 4.1. Let u 0 ∈ H s+2 (R) and v 0 ∈ H s+1 (R) for some s > 1/2. Then there
exists a T > 0, depending only upon of (u 0 , v 0 ) ∈ H s+2 (R) × H s+1 (R), and a unique
function u ∈ Y s (T ) which is solution of the equation in (1.1) in the distributional sense on
R × [0, T ], and for which u(·, 0) = u 0 and u t (·, 0) = v 0 ′ . The solution depends continuously
upon the data (u 0 , v 0 ) in the sense that the mapping that associates to (u 0 , v 0 ) the solution
u is continuous from H s+2 (R) × H s+1 (R) into Y s (T ). If s > 5/2 the solution is classical.
Proof. See Corollary 2 in [6].
□
Now, consider the equation
u tt − u xx + u xxxx + (|u| α−1 u) xx = 0. (4.1)
Suppose that u is a solution of the initial-value problem (1.1) given by Theorem 4.1
for s sufficiently large, thus we can proceed as follows.
Apply the operarator (−∆) −1 to the equation (4.1) and multiply by u t , then integrating
respect to x, we obtain the following
or
1 d {
‖(−∆) −1/2 u t ‖ 2 2 + ‖u‖ 2 2 + ‖u x ‖ 2 2 − 2
2 dt
α + 1
α+1} ‖u‖α+1 = 0
‖(−∆) −1/2 u t ‖ 2 2 + ‖u‖ 2 2 + ‖u x ‖ 2 2 − 2
α + 1 ‖u‖α+1
= K 0 (4.2)
where K 0 = K 0 (f 0 , g 0 ).
The relation (4.2) is the main tool to show that ‖u(t)‖ 1,2 remains bounded on the interval
[0, T ′ ) for small data, and so ‖(−∆) −1/2 u t ‖ 2 does. Hence we can apply the Theorem
4.2 again to continue the solution. So we have the following
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