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A note on bilinear estimates and regularity of flow maps for nonlinear dispersive equations

Author(s): Sebastian Herr
Journal: Proc. Amer. Math. Soc. 136 (2008), 2881-2886.
MSC (2000): Primary 35Q53; Secondary 76B15, 35B30
Posted: April 8, 2008
MathSciNet review: 2399054
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Abstract | References | Similar articles | Additional information

Abstract: Explicit counterexamples to bilinear estimates related to the Benjamin-Ono equation in the periodic setting are calculated for functions of zero mean value. As a consequence, certain bilinear estimates fail to hold in spite of the analyticity of the flow map. The latter has been shown recently by L. Molinet.

References:

1.: Jean Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3 (1993), no. 3, 209-262. MR 1215780 (95d:35160b)
2.: Sebastian Herr, Well-posedness for equations of Benjamin-Ono type,
Illinois J. Math. 51 (2007), no. 3, 951-976.
3.: Alexandru D. Ionescu and Carlos E. Kenig, Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, J. Amer. Math. Soc. 20 (2007), 753-798. MR 2291918
4.: -, Complex-valued solutions of the Benjamin-Ono equation. In Harmonic Analysis, Partial Differential Equations, and Related Topics, Contemp. Math., vol. 428, Amer. Math. Soc., Providence, RI, 2007, pp. 61-74. MR 2322378
5.: Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), no. 4, 527-620. MR 1211741 (94h:35229)
6.: -, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), no. 2, 573-603. MR 1329387 (96k:35159)
7.: Herbert Koch and Nikolay Tzvetkov, Nonlinear wave interactions for the Benjamin-Ono equation, Int. Math. Res. Not. (2005), no. 30, 1833-1847. MR 2172940 (2006f:35245)
8.: Luc Molinet, Jean-Claude Saut, and Nikolay Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal. 33 (2001), no. 4, 982-988 (electronic). MR 1885293 (2002k:35281)
9.: Luc Molinet, Global well-posedness in the energy space for the Benjamin-Ono equation on the circle, Math. Ann. 337 (2007), no. 2, 353-383. MR 2262788
10.: -,
Global well-posedness in for the periodic Benjamin-Ono equation, to appear in Amer. J. Math., arXiv:math.AP/0601217
11.: Terence Tao, Global well-posedness of the Benjamin-Ono equation in $H\sp 1({\bf R})$ , J. Hyperbolic Differ. Equ. 1 (2004), no. 1, 27-49. MR 2052470 (2005f:35273)

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Additional Information:

Sebastian Herr
Affiliation: Technische Universität Dortmund, Fakultät für Mathematik, 44221 Dortmund, Germany
Address at time of publication: Center for Pure and Applied Mathematics, University of California, 837 Evans Hall, Berkeley, California 94720-3840
Email: herr@math.berkeley.edu

DOI: 10.1090/S0002-9939-08-09238-1
PII: S 0002-9939(08)09238-1
Keywords: Failure of bilinear estimates, periodic Benjamin-Ono equation
Received by editor(s): April 17, 2007
Posted: April 8, 2008
Additional Notes: The author is grateful to M. Hadac, H. Koch, and N. Tzvetkov for valuable discussions on the subject. This research is part of the author's doctoral dissertation, which has been awarded a Dissertationspreis at Dortmund University
Communicated by: Hart F. Smith
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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