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Mathematische Annalen

, Volume 345, Issue 2, pp 307–366 | Cite as

Rough blowup solutions to the L 2 critical NLS

  • James CollianderEmail author
  • Pierre Raphaël
Article

Abstract

We study the singularity formation for the cubic focusing L 2-critical nonlinear Schrödinger equation on \({\mathbb{R}^{2}}\) . In a series of recent works, Merle and Raphaël have completely described the so called log–log blowup regime and proven its stability in the energy space H 1. Our aim in this paper is to investigate the stability of this blowup regime under rough perturbations in the direction of developing a theory at the level of the critical space L 2. By blending the Merle, Raphaël techniques with the quantitative I-method developed by Colliander, Keel, Staffilani, Takaoka and Tao for the study of the Cauchy problem for rough data, we obtain the stability of the log–log regime in H s for all s > 0.

Mathematics Subject Classification (2000)

35Q55 37K40 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Institut de MathématiquesUniversité Paul SabatierToulouseFrance

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