Electron. J. Diff. Eqns., Vol. 2009(2009), No. 52, pp. 1-48.

Diophantine conditions in global well-posedness for coupled KdV-type systems

Tadahiro Oh

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.

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Tadahiro Oh
Department of Mathematics, University of Toronto
40 St. George St, Rm 6290
Toronto, ON M5S 2E4, Canada
email: oh@math.toronto.edu

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