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International Mathematics Research Papers (2006) Vol. 2006 : article ID 70630, 84 pages, doi:10.1155/IMRP/2006/70630
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Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

Quadratic nonlinear derivative Schrödinger equations. Part I

Ioan Bejenaru

Department of Mathematics, University of California, Los Angeles (UCLA) Los Angeles, CA 90095-1555, USA E-mail address: bejenaru{at}math.ucla.edu

We consider the local well-posedness theory for the quadratic nonlinear Schrödinger equation with low regularity initial data in the case when the nonlinearity contains derivatives. We work in 2 + 1 dimensions and prove a local well-posedness result up to the scaling for small initial data with some spherical symmetry structure.



References

  1. Chihara H. Gain of regularity for semilinear Schrödinger equations. Mathematische Annalen (1999) 315(4):529–567.[CrossRef][Web of Science]
  2. Christ M. Illposedness of a Schrödinger equation with derivative nonlinearity. preprint, http://math.berkeley.edu/~mchrist/preprints.html.
  3. Colliander J. E., Delort J.-M., Kenig C. E., Staffilani G. Bilinear estimates and applications to 2D NLS. Transactions of the American Mathematical Society (2001) 353(8):3307–3325.[CrossRef][Web of Science]
  4. Gruenrock A. On the Cauchy- and periodic boundary value problem for a certain class of derivative nonlinear Schrödinger equations. preprint, http://xxx.lanl.gov/abs/math.AP/0006195.
  5. Hörmander L. The Analysis of Linear Partial Differential Operators. III. In: Fundamental Principles of Mathematical Sciences (1985) 274. Berlin: Springer. viii+525. chapter 22.
  6. Kenig C. E., Ponce G., Vega L. Small solutions to nonlinear Schrödinger equations. Annales de l'Institut Henri Poincaré. Analyse Non Linéaire (1993) 10(3):255–288.
  7. Kenig C. E., Ponce G., Vega L. Quadratic forms for the 1-D semilinear Schrödinger equation. Transactions of the American Mathematical Society (1996) 348(8):3323–3353.[CrossRef][Web of Science]
  8. Kenig C. E., Ponce G., Vega L. On the Cauchy problem for linear Schrödinger systems with variable coefficient lower order terms. In: CMS Conference Proceedings (1997) 21. Harmonic Analysis and Number Theory, 1996: Montreal, PQ. Rhode Island: American Mathematical Society. 205–227.
  9. Kenig C. E., Ponce G., Vega L. On the smoothing properties of some dispersive hyperbolic systems. In: GAKUTO International Series. Mathematical Sciences and Applications (1997) 10. Nonlinear Waves, 1995: Sapporo. Tokyo: Gakkotosho. 221–229.
  10. Kenig C. E., Ponce G., Vega L. Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations. Inventiones Mathematicae (1998) 134(3):489–545.[CrossRef][Web of Science]
  11. Mizohata S. On the Cauchy Problem. In: Notes and Reports in Mathematics in Science and Engineering (1985) 3. Florida; Beijing: Academic Press; Science Press. vi+177.
  12. Nahmod A., Stefanov A., Uhlenbeck K. On Schrödinger maps. Communications on Pure and Applied Mathematics (2003) 56(1):114–151.[CrossRef][Web of Science]
  13. Tao T. Global regularity of wave maps II. Small energy in two dimensions. Communications in Mathematical Physics (2001) 224(2):443–544.[CrossRef][Web of Science]
  14. Tataru D. Local and global results for wave maps I. Communications in Partial Differential Equations (1998) 23(9-10):1781–1793.[CrossRef][Web of Science]
  15. Tataru D. Rough solutions for the Wave-Maps equation. American Journal of Mathematics (2005) 127(2):293–377.[CrossRef][Web of Science]

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This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
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