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Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line
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Choffrut, Antoine and Pocovnicu, Oana (2017) Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line. International Mathematics Research Notes, 2018 (3). pp. 699-738. doi:10.1093/imrn/rnw246
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Official URL: https://doi.org/10.1093/imrn/rnw246
Abstract
In this paper, we study ill-posedness of cubic fractional nonlinear Schrödinger equations. First, we consider the cubic nonlinear half-wave equation (NHW) on R. In particular, we prove the following ill-posedness results: (i) failure of local uniform continuity of the solution map in Hs(R) for s∈(0,1/2), and also for s=0 in the focusing case; (ii) failure of C3- smoothness of the solution map in L2(R); (iii) norm inflation and, in particular, failure of continuity of the solution map in Hs(R), s<0. By a similar argument, we also prove norm inflation in negative Sobolev spaces for the cubic fractional NLS. Surprisingly, we obtain norm inflation above the scaling critical regularity in the case of dispersion |D|β with β>2.
| Item Type: | Journal Article | ||||||||
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| Subjects: | Q Science > QC Physics | ||||||||
| Divisions: | Faculty of Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Schrödinger equation | ||||||||
| Journal or Publication Title: | International Mathematics Research Notes | ||||||||
| Publisher: | Oxford University Press | ||||||||
| ISSN: | 1073-7928 | ||||||||
| Official Date: | 31 January 2017 | ||||||||
| Dates: |
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| Volume: | 2018 | ||||||||
| Number: | 3 | ||||||||
| Page Range: | pp. 699-738 | ||||||||
| DOI: | 10.1093/imrn/rnw246 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Funder: | University of Edinburgh. Whittaker Research Fellowship, European Research Council (ERC), National Science Foundation (U.S.) (NSF) | ||||||||
| Grant number: | Grant agreement no. 616797 (ERC), Grant no. DMS-1440140 (NSF) | ||||||||
| Open Access Version: |
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