Rough blowup solutions to the L2 critical NLS

We study the singularity formation for the cubic focusing L-2-critical nonlinear Schrodinger equation on 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 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, 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 H-s for all s > 0.