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

Colliander, J.¹ ; Keel, M.² ; Staffilani, G.^3, 4 ; Takaoka, H.^5, 6 ; Tao, T.⁷

¹ Department of Mathematics, University of Toronto, Toronto, ON Canada, M5S 3G3
² School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455
³ Department of Mathematics, Stanford University, Stanford, California 94305-2125
⁴ Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02138
⁵ Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
⁶ Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan
⁷ Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095-1555

Journal of the American Mathematical Society, Tome 16 (2003) no. 3, pp. 705-749

Résumé

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.

Détail

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

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