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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

**1.**I. Bejenaru, T. Tao,*Sharp well-posedness and ill-posedness results for a quadratic nonlinear Schrödinger equation*, J. of Func. Anal. vol. 233, issue 1, pp. 228-259. MR**2204680 (2007i:35216)****2.**I. Bejenaru,*Quadratic Nonlinear Derivative Schrödinger Equations - Part 2*, preprint available on arxiv.**3.**B. Birnir, C. Kenig, G. Ponce, N. Svanstedt, L. Vega,*On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations*, J. London Math. Soc. (2)**53**(1996), 551-559. MR**1396718 (97d:35233)****4.**J. Bourgain,*Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations*, Part I, Geometric and Funct. Anal.**3**(1993), 107-156. MR**1209299 (95d:35160a)****5.**T. Cazenave, F.B. Weissler,*The Cauchy problem for the critical nonlinear Schrödinger equation*, Non. Anal. TMA**14**(1990), 807-836. MR**1055532 (91j:35252)****6.**J. Colliander, J. Delort, C. Kenig, G. Staffilani,*Bilinear Estimates and Applications to 2D NLS*, Trans. Amer. Math. Soc.**353**(2001), no. 8, 3307-3325. MR**1828607 (2002d:35186)****7.**M. Christ, J. Colliander, T. Tao,*Low-regularity ill-posedness for nonlinear Schrödinger and wave equations*, preprint.**8.**C. Kenig, G. Ponce, L. Vega,*Quadratic forms for the 1-D semilinear Schrödinger equation*, Trans. Amer. Math. Soc.**346**(1996), 3323-3353. MR**1357398 (96j:35233)****9.**C. Kenig, G. Ponce, L. Vega,*On the ill-posedness of some canonical dispersive equations*, Duke Math. J.**106**(2001), 617-633. MR**1813239 (2002c:35265)****10.**T. Muramatu, S. Taoka,*The initial value problem for the 1-D semilinear Schrödinger equation in Besov spaces*, J. Math. Soc. Japan**56**(2004), no. 3, 853-888. MR**2071676 (2005h:35325)****11.**K. Nakanishi, H. Takaoka, Y. Tsutsumi,*Counterexamples to bilinear estimates related with the KdV equation and the nonlinear Schrödinger equation*, Methods and Applications of Analysis**8**(2001), 569-578. MR**1944182 (2004c:35364)****12.**D. Tataru,*Local and global results for wave maps I*, Comm. PDE**23**(1998), 1781-1793. MR**1641721 (99j:58209)****13.**T. Tao,*Multilinear weighted convolution of functions, and applications to nonlinear dispersive equations*, Amer. J. Math.**123**(2001), no. 5, 839-908. MR**1854113 (2002k:35283)****14.**Y. Tsutsumi,*solutions for nonlinear Schrodinger equations and nonlinear groups*, Funk. Ekva.**30**(1987), 115-125. MR**915266 (89c:35143)**

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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