Well-posedness for the fifth-order KdV equation in the energy space
Abstract
We prove that the initial value problem (IVP) associated to the fifth order KdV equation {equation} \label{05KdV} \partial_tu-\alpha\partial^5_x u=c_1\partial_xu\partial_x^2u+c_2\partial_x(u\partial_x^2u)+c_3\partial_x(u^3), {equation} where
x∈R
,
t∈R
,
u=u(x,t)
is a real-valued function and
α,
c
1
,
c
2
,
c
3
are real constants with
α≠0
, is locally well-posed in
H
s
(R)
for
s≥2
. In the Hamiltonian case (\textit i.e. when
c
1
=
c
2
), the IVP associated to
(???)
is then globally well-posed in the energy space
H
2
(R)
.