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Transactions of the American Mathematical Society

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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 | References | Similar Articles | Additional Information


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

J.-M. Delort
Affiliation: Département of Mathématiques, Université de Paris-Nord, 93430 Villetaneuse, France

C. E. Kenig
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637

G. Staffilani
Affiliation: Department of Mathematics, Stanford University, Stanford California 94305

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

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