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Rough blowup solutions to the L 2 critical NLS

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada

    James Colliander

  2. Institut de Mathématiques, Université Paul Sabatier, 31062, Toulouse, France

    Pierre Raphaël

Authors
  1. James Colliander
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  2. Pierre Raphaël
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Correspondence to James Colliander.

Additional information

J. Colliander is supported in part by N.S.E.R.C. Grant R.G.P.I.N. 250233-07 and P. Raphaël was supported in part by the Agence Nationale de la Recherche, ANR ONDENONLIN.

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Colliander, J., Raphaël, P. Rough blowup solutions to the L 2 critical NLS. Math. Ann. 345, 307–366 (2009). https://doi.org/10.1007/s00208-009-0355-3

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  • DOI: https://doi.org/10.1007/s00208-009-0355-3

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