877 research outputs found
Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line
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 . In particular, we prove the following
ill-posedness results: (i) failure of local uniform continuity of the solution
map in for , and also for in the
focusing case; (ii) failure of -smoothness of the solution map in
; (iii) norm inflation and, in particular, failure of
continuity of the solution map in , . 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 with .Comment: Introduction expanded, references updated. We would like to thank
Nobu Kishimoto for his comments on the previous version and for pointing out
the related article of Iwabuchi and Uriy
Numerical simulations for the energy-supercritical nonlinear wave equation
We carry out numerical simulations of the defocusing energy-supercritical
nonlinear wave equation for a range of spherically-symmetric initial
conditions. We demonstrate numerically that the critical Sobolev norm of
solutions remains bounded in time. This lends support to conditional scattering
results that have been recently established for nonlinear wave equations.Comment: 28 pages, 13 figures. New references and new cases adde
The defocusing energy-supercritical NLS in four space dimensions
We consider a class of defocusing energy-supercritical nonlinear
Schr\"odinger equations in four space dimensions. Following a
concentration-compactness approach, we show that for , any solution
that remains bounded in the critical Sobolev space must
be global and scatter. Key ingredients in the proof include a long-time
Strichartz estimate and a frequency-localized interaction Morawetz inequality.Comment: 52 page
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