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

Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T.

doi:10.1090/S0894-0347-03-00421-1

Sharp global well-posedness for KdV and modified KdV on $\mathbb R$ and $\mathbb T$
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by J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao

J. Amer. Math. Soc. 16 (2003), 705-749

DOI: https://doi.org/10.1090/S0894-0347-03-00421-1

Published electronically: January 29, 2003

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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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