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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

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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
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:
DateEvent
31 January 2017Published
24 December 2016Available
13 October 2016Accepted
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)
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