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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Low regularity solutions for a 2D quadratic nonlinear Schrödinger equation


Authors: Ioan Bejenaru and Daniela De Silva
Journal: Trans. Amer. Math. Soc. 360 (2008), 5805-5830
MSC (2000): Primary 35Q55
DOI: https://doi.org/10.1090/S0002-9947-08-04415-2
Published electronically: June 19, 2008
MathSciNet review: 2425697
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Abstract | References | Similar Articles | Additional Information

Abstract: We establish that the initial value problem for the quadratic non-linear Schrödinger equation

$\displaystyle iu_t - \Delta u = u^2,$

where $ u: \mathbb{R}^2 \times \mathbb{R} \to \mathbb{C}$, is locally well-posed in $ H^s(\mathbb{R}^2)$ when $ s > -1$. The critical exponent for this problem is $ s_c=-1$, and previous work by Colliander, Delort, Kenig and Staffilani, 2001, established local well-posedness for $ s > -3/4$.


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Additional Information

Ioan Bejenaru
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095
Address at time of publication: Department of Mathematics, Texas A & M University, College Station, Texas 77843

Daniela De Silva
Affiliation: Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218
Address at time of publication: Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10027

DOI: https://doi.org/10.1090/S0002-9947-08-04415-2
Received by editor(s): August 21, 2006
Published electronically: June 19, 2008
Additional Notes: The authors were partially supported by the Mathematical Sciences Research Institute (MSRI) at Berkeley.
Article copyright: © Copyright 2008 American Mathematical Society

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