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 , we prove the global well-posedness in for and for via the I-method developed by Colliander-Keel-Staffilani-Takaoka-Tao . When , as in the local theory , certain resonances occur, closely depending on the value of . 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 , , where is determined by the Diophantine characterization of certain constants derived from the coupling parameter . 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
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Toronto, ON M5S 2E4, Canada
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