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

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J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao
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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.## References

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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
- MR Author ID: 614986
**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
- MR Author ID: 361755
- ORCID: 0000-0002-0140-7641
- 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 - © Copyright 2003 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**16**(2003), 705-749 - MSC (2000): Primary 35Q53, 42B35, 37K10
- DOI: https://doi.org/10.1090/S0894-0347-03-00421-1
- MathSciNet review: 1969209