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

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 L^2 -based Sobolev spaces H s H^s where local well-posedness is presently known, apart from the H 1 4 ( R ) H^{\frac {1}{4}} (\mathbb {R} ) endpoint for mKdV and the H − 3 4 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.