Transactions of the American Mathematical Society

Author(s): J. E. Colliander; J.-M. Delort; C. E. Kenig; G. Staffilani
Journal: Trans. Amer. Math. Soc. 353 (2001), 3307-3325.
MSC (2000): Primary 35Q55, 42B35
Posted: April 10, 2001
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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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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: 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
Posted: 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
Copyright of article: Copyright 2001, American Mathematical Society

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