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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations


Authors: H. A. Biagioni and F. Linares
Journal: Trans. Amer. Math. Soc. 353 (2001), 3649-3659
MSC (1991): Primary 35Q55, 35Q51
Published electronically: May 3, 2001
MathSciNet review: 1837253
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Abstract:

Ill-posedness is established for the initial value problem (IVP) associated to the derivative nonlinear Schrödinger equation for data in $H^s(\mathbb R)$, $s<1/2$. This result implies that best result concerning local well-posedness for the IVP is in $H^s(\mathbb R),\, s\ge1/2$. It is also shown that the (IVP) associated to the generalized Benjamin-Ono equation for data below the scaling is in fact ill-posed.


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Additional Information

H. A. Biagioni
Affiliation: Departamento de Matemática, IMECC-UNICAMP, 13081-970, Campinas, SP, Brasil
Email: hebe@ime.unicamp.br

F. Linares
Affiliation: Instituto de Matemática Pura e Aplicada, 22460-320, Rio de Janeiro, Brasil
Email: linares@impa.br

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02754-4
PII: S 0002-9947(01)02754-4
Keywords: Ill-posedness, Schr\"odinger equation, Benjamin-Ono equation
Received by editor(s): April 5, 2000
Received by editor(s) in revised form: July 24, 2000
Published electronically: May 3, 2001
Article copyright: © Copyright 2001 American Mathematical Society