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Sharp global well-posedness for KdV and modified KdV on \(\mathbb R\) and \(\mathbb T\). (English) Zbl 1025.35025

The authors consider the \(\mathbb R \)-valued Korteweg-de Vries (KdV) equation \[ u_t+u_{xxx}+uu_x =0,\quad u:{\mathbb R}\times[0,T]\mapsto {\mathbb R}. \] They prove the global well-posedness (GWP) of KdV for initial data in \(H^s({\mathbb R})\), \(s>-3/4\). The local well-posedness of KdV in \(H^s\) for \(s>-3/4\) was obtained by C. E. Kenig, G. Ponce and L. Vega [J. Am. Math. Soc. 9, 573-603 (1996; Zbl 0848.35114)].
The range of \(s\) is sharp; M. Christ, J. Colliander and T. Tao [Am. J. Math. 125, No. 6, 1235–1293 (2003; Zbl 1048.35101)] proved the ill-posedness of KdV for \(s<-3/4\) in the sense that the solution operator is not uniformly continuous with respect to the \(H^s\) norm.
The authors also prove various GWP results for periodic KdV and modified KdV equations.
The paper is well written and highly readable. The introduction includes a detailed history of related results as well as a useful heuristic discussion of their method.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
42B35 Function spaces arising in harmonic analysis
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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