Bilinear estimates and applications to 2d NLS

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 fo...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2001-08, Vol.353 (8), p.3307-3325
Main Authors: Colliander, J. E., Delort, J.-M., Kenig, C. E., Staffilani, G.
Format: Article
Language:English
Subjects:
A priori knowledge
Cauchy problem
Cosine function
Cubes
Dyadics
Fourier transformations
Low frequencies
Mathematical functions
Preliminary estimates
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Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-01-02760-X