Low regularity blowup solutions for the mass-critical NLS in higher dimensions

In this paper, we study the $H^s$-stability of the log-log blowup regime (which has been completely described in a series of recent works by Merle and Raphael) for the focusing mass-critical nonlinear Schr"odinger equations $i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u+\mathrm{\mid }u{\mathrm{\mid }}^{\frac{4}{d}}u=0$ in ${\mathbb{R}}^d$ with $d\ge 3$. We aim to extend the result in [Colliander and Raphael, Rough blowup solutions to the $L^2$ critical NLS, Math. Anna., 345(2009), 307-366.] for dimension two to the higher dimensions cases $d\ge 3$, where we use the bootstrap argument in the above paper and the commutator estimates in [M. Visan and X. Zhang, On the blowup for the $L^2$-critical focusing nonlinear Schr"odinger equation in higher dimensions below the energy class. SIAM J. Math. Anal., 39(2007), 34-56.].