Global well-posedness for KdV in Sobolev spaces of negative index

By James E. Colliander, Markus Keel, Gigliola Staffilani, Hideo Takaoka, T. Tao. 2001

102 citations recorded by the source corpus.

The initial value problem for the Korteweg-deVries equation on the line is shown to be globally well-posed for rough data. In particular, we show global well-posedness for initial data in $H^s(\mathbb{R})$ for -3/10 < s.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9070d7757535dfbdf4b63889297d3b41809b6bbd

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In Mathematical Physics

Citations in the corpus

On global solutions to a defocusing semi-linear wave equation (2003, background): “Indeed, in all the recent progress for other (technically more complicated) equations like KdV ([6]) or NLS ([5]) the same kind of restrictions appear, through for KdV it solves completely the well-posedness question as the equation is ill-posed below half-way to the scaling ([12]).”
Sharp Global well-posedness for KdV and modified KdV on $\R$ and $\T$ (2001, unknown): “We showed that the low frequency issue may indeed be circumvented in [12] by proving (1.”
Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ℝ3 (2003, background): “See [11, 12, 21, 23] for further applications of almost conservation laws, and [14, 16, 17] for instances where the inclusion of correction terms in the almost conserved energy leads to sharp results.”
Sharp global well-posedness for KdV and modified KdV on ℝ and (2003, background): “[18], [ 15 ]. The estimate (1.3) fails for s 15 ] by proving (1.1) is GWP inHs(R) ;s > 3 10 . The approach in [15] does not rely on showing the nonlinear Duhamel term has regularity at the level of the conservation law.”
Multilinear estimates for periodic KdV equations, and applications (2001, methodology): “We prove this in Sections 11-13, by using the “I-method” in [21], [9], [ 10 ] (see also [20], [11]); this is a method for obtaining global well-posedness below the energy norm by constructing “almost conserved” analogues of the Hamiltonian for rough solutions.”