Blow-up solutions on a sphere for the 3d quintic NLS in the energy space

We prove that if $u(t)$ is a log-log blow-up solution, of the type studied by Merle-Rapha"el (2001-2005), to the $L^2$ critical focusing NLS equation $i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u+\mathrm{\mid }u{\mathrm{\mid }}^{4\mathrm{/}d}u=0$ with initial data $u_0\in H^1($\mathbb{R}$^d)$inthecases$d=1, 2$,then$u(t)$remainsboundedin$H^1$ away from the blow-up point. This is obtained without assuming that the initial data $u_0$ has any regularity beyond $H^1({\mathbb{R}}^d)$. As an application of the $d=1$ result, we construct an open subset of initial data in the radial energy space $H_{rad}^1({\mathbb{R}}^3)$ with corresponding solutions that blow-up on a sphere at positive radius for the 3d quintic (${\dot{H}}^1$-critical) focusing NLS equation $i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u+\mathrm{\mid }u{\mathrm{\mid }}^{4}u=0$. This improves Rapha"el-Szeftel (2009), where an open subset in $H_{rad}^3({\mathbb{R}}^3)$ is obtained. The method of proof can be summarized as follows: on the whole space, high frequencies above the blow-up scale are controlled by the bilinear Strichartz estimates. On the other hand, outside the blow-up core, low frequencies are controlled by finite speed of propagation.