Rough blowup solutions to the L 2 critical NLS

Abstract

We study the singularity formation for the cubic focusing L2-critical nonlinear Schrodinger equation on \({\mathbb{R}^{2}}\) . In a series of recent works, Merle and Raphael have completely described the so called log–log blowup regime and proven its stability in the energy space H1. 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 L2. By blending the Merle, Raphael 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 Hs for all s > 0.