Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line

Antoine Choffrut, Oana Pocovnicu

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Abstract

In this paper, we study ill-posedness of cubic fractional nonlinear Schr\"odinger equations. First, we consider the cubic nonlinear half-wave equation (NHW) on $\mathbb R$. In particular, we prove the following ill-posedness results: (i) failure of local uniform continuity of the solution map in $H^s(\mathbb R)$ for $s\in (0,\frac 12)$, and also for $s=0$ in the focusing case; (ii) failure of $C^3$-smoothness of the solution map in $L^2(\mathbb R)$; (iii) norm inflation and, in particular, failure of continuity of the solution map in $H^s(\mathbb R)$, $s 2$.
Original languageEnglish
Article numberrnw246
Number of pages41
JournalInternational Mathematics Research Notices
Volume2016
DOIs
Publication statusPublished - 24 Dec 2016

Keywords

  • math.AP

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