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

DOI:
https://doi.org/10.1090/S0002-9947-01-02760-X

Published electronically:
April 10, 2001

MathSciNet review:
1828607

Full-text PDF

Abstract | References | Similar Articles | Additional Information

The three bilinearities for functions are sharply estimated in function spaces associated to the Schrödinger operator . 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:
https://doi.org/10.1090/S0002-9947-01-02760-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