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Sharp global well-posedness for KdV and modified KdV on $\Bbb R$ and $\Bbb T$. (English) Zbl 1025.35025
The authors consider the $\Bbb R $-valued Korteweg-de Vries (KdV) equation $$ u_t+u_{xxx}+uu_x =0,\quad u:{\Bbb R}\times[0,T]\mapsto {\Bbb R}. $$ They prove the global well-posedness (GWP) of KdV for initial data in $H^s({\Bbb R})$, $s>-3/4$. The local well-posedness of KdV in $H^s$ for $s>-3/4$ was obtained by {\it C. E. Kenig, G. Ponce} and {\it L. Vega} [J. Am. Math. Soc. 9, 573-603 (1996; Zbl 0848.35114)]. The range of $s$ is sharp; {\it M. Christ, J. Colliander} and {\it 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.

35Q53KdV-like (Korteweg-de Vries) equations
42B35Function spaces arising in harmonic analysis
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
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