Notes on Boussinesq Equation

4.2. PERSISTENCE 41
We begin by stating the following theorem that deals with the persistence properties
in H s for solutions of the IVP (4.4). In this section we use g to denote g = h ′ .
Theorem 4.3. Let (f, g) ∈ H s+1 (R) × H s (R), s ≥ 1, with ‖f‖ 1,2 and ‖g‖ 2 small, then
there exists a unique solution u ∈ C(R + : H 1 (R)) of the IVP (4.4) such that
u ∈ C b (R + : H 1 (R)) ∩ C(R + : H s+1 (R)).
Proof. Applying ∂x
α to the equation in (4.4), multiplying by ∂x α u t , and integrating
with respect to x, and then integrating by parts we have
∫
1 d {
|∂x α u t | 2 + |∂x α u x | 2 + |∂x α u xx | 2} ∫
dx = ∂x α (u 2 ) xx ∂x α u t dx.
2 dt
Now, using the Cauchy-Schwarz and Gagliardo-Nirenberg inequalities we have
{
}
d
‖∂x α u t ‖ 2 2 + ‖∂x α u x ‖ 2 2 + ‖∂x α u xx ‖ 2 2 ≤ c ‖u‖ ∞ ‖∂x α u xx ‖ 2 ‖∂x α u t ‖ 2 .
dt
Gronwall’s inequality gives
(
‖∂
α
x u t ‖ 2 + ‖∂x α u x ‖ 2 + ‖∂x α )
u xx ‖ 2
sup
[0,T ]
≤
} ( ∫ {‖∂ T
x α g‖ 2 + ‖∂x α f x ‖ 2 + ‖∂x α f xx ‖ 2 exp C
0
)
‖u(t)‖ ∞ dt .
As usual the above formal computation can be justified by using the continuous dependence
on the data.
Now, using Theorem 4.2 the result follows.
□
Theorem 4.4. Consider the IVP (4.4) with f ∈ H 3 (R), g ∈ H 2 (R). If xf xx , xf x and
xg ∈ L 2 (R), then there exists T > 0 such that
xu ∈ C([0, T ] : H 2 (R)).
Moreover, if ‖f‖ 1,2 , ‖g‖ 2 ≪ 1, the result is global, i.e.
Proof. From (4.4) we have
xu ∈ C(R + : H 2 (R)).
(xu) tt = (xu) xx − (xu) xxxx − 2(xu) x u x − 2(xu) xx u − 2u x + 4u xxx + 6uu x
multiplying by (xu) t and integrating with respect to x, we find
1 d
{ ∫ ∫
∫
}
[(xu) t ] 2 dx + [(xu) x ] 2 dx + [(xu) xx ] 2 dx
2 dt
∫
∫
= −2 u x (xu) x (xu) t dx − 2 u(xu) xx (xu) t dx.
∫
∫
∫
+ 4 u xxx (xu) t dx − 2 u x (xu) t dx + 6 uu x (xu) t