Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 52, pp. 1-48.
Title: Diophantine conditions in global well-posedness for coupled
KdV-type systems
Author: Tadahiro Oh (Univ. of Toronto, Canada)
Abstract:
We consider the global well-posedness problem of a one-parameter
family of coupled KdV-type systems both in the periodic and non-periodic
setting. When the coupling parameter $\alpha = 1$, we prove the global
well-posedness in $H^s(\mathbb{R}) $ for $s > 3/4$
and $H^s(\mathbb{T}) $ for $s \geq -1/2$ via the I-method developed
by Colliander-Keel-Staffilani-Takaoka-Tao [5].
When $\alpha \ne 1$, as in the local theory [14],
certain resonances occur, closely depending on the value of $\alpha$.
We use the Diophantine conditions to characterize the resonances.
Then, via the second iteration of the I-method, we establish a
global well-posedness result in $H^s(\mathbb{T})$, $s \geq \widetilde{s}$,
where $\widetilde{s}= \widetilde{s}(\alpha) \in (5/7, 1]$ is determined by the
Diophantine characterization of certain constants derived from the
coupling parameter $\alpha$. We also show that the third iteration of
the I-method fails in this case.
Submitted August 2, 2008. Published April 14, 2009.
Math Subject Classifications: 35Q53.
Key Words: KdV; global well-posedness; I-method; Diophantine condition.