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Sharp global well-posedness for KdV and modified KdV on $\mathbb R$ and $\mathbb T$

Authors: J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao
Journal: J. Amer. Math. Soc. 16 (2003), 705-749
MSC (2000): Primary 35Q53, 42B35, 37K10
Published electronically: January 29, 2003
MathSciNet review: 1969209
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Abstract: The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all $L^2$-based Sobolev spaces $H^s$ where local well-posedness is presently known, apart from the $H^{\frac{1}{4}} ({\mathbb R})$ endpoint for mKdV and the $H^{-\frac{3}{4}}$endpoint for KdV. The result for KdV relies on a new method for constructing almost conserved quantities using multilinear harmonic analysis and the available local-in-time theory. Miura's transformation is used to show that global well-posedness of modified KdV is implied by global well-posedness of the standard KdV equation.

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Additional Information

J. Colliander
Affiliation: Department of Mathematics, University of Toronto, Toronto, ON Canada, M5S 3G3

M. Keel
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455

G. Staffilani
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305-2125
Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02138

H. Takaoka
Affiliation: Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
Address at time of publication: Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan

T. Tao
Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095-1555

Keywords: Korteweg-de Vries equation, nonlinear dispersive equations, bilinear estimates, multilinear harmonic analysis
Received by editor(s): November 7, 2001
Received by editor(s) in revised form: November 19, 2002
Published electronically: January 29, 2003
Additional Notes: The first author is supported in part by N.S.F. Grant DMS 0100595.
The second author is supported in part by N.S.F. Grant DMS 9801558
The third author is supported in part by N.S.F. Grant DMS 9800879 and by a grant from the Sloan Foundation
The fourth author is supported in part by J.S.P.S. Grant No. 13740087
The last author is a Clay Prize Fellow and is supported in part by grants from the Packard and Sloan Foundations
Article copyright: © Copyright 2003 American Mathematical Society