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

 

Bilinear estimates and applications to 2d NLS


Authors: J. E. Colliander, J.-M. Delort, C. E. Kenig and G. Staffilani
Journal: Trans. Amer. Math. Soc. 353 (2001), 3307-3325
MSC (2000): Primary 35Q55, 42B35
Published electronically: April 10, 2001
MathSciNet review: 1828607
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Abstract:

The three bilinearities $u v, \overline{uv},\overline{u}v$ for functions $u, v : \mathbb{R} ^2 \times [0,T] \longmapsto \mathbb{C} $ are sharply estimated in function spaces $X_{s,b}$ associated to the Schrödinger operator $i \partial_t + \Delta $. These bilinear estimates imply local wellposedness results for Schrödinger equations with quadratic nonlinearity. Improved bounds on the growth of spatial Sobolev norms of finite energy global-in-time and blow-up solutions of the cubic nonlinear Schrödinger equation (and certain generalizations) are also obtained.


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Additional Information

J. E. Colliander
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Email: colliand@math.berkeley.edu

J.-M. Delort
Affiliation: Département of Mathématiques, Université de Paris-Nord, 93430 Villetaneuse, France
Email: delort@math.univ-paris13.fr

C. E. Kenig
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: cek@math.uchicago.edu

G. Staffilani
Affiliation: Department of Mathematics, Stanford University, Stanford California 94305
Email: gigliola@math.stanford.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02760-X
PII: S 0002-9947(01)02760-X
Keywords: Nonlinear Schr\"odinger equation, nonlinear dispersive equations, weak turbulence, NLS blow-up, bilinear estimates, multilinear harmonic analysis, Strichartz inequalities
Received by editor(s): July 24, 2000
Published electronically: April 10, 2001
Additional Notes: J.E.C. was supported in part by an N.S.F. Postdoctoral Research Fellowship.
C.E.K. was supported in part by N.S.F. Grant DMS 9500725
G.S. was supported in part by N.S.F. Grant DMS 9800879
Article copyright: © Copyright 2001 American Mathematical Society