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 and Raphal, to the L 2 critical focusing NLS equation i t u + u + |u| 4/d u = 0 with initial data u 0 H 1 ( d ) in the cases d = 1, 2, then u(t) remains bounded in 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 ( d ). As an application of the d = 1 result, we construct an open subset of initial data in the radial energy space H 1 rad ( 3 ) with corresponding solutions that blow up on a sphere at positive radius for the 3D quintic ( 1 -critical) focusing NLS equation i t u + u + |u| 4 u = 0. This improves the results of Raphal and Szeftel 2009, where an open subset in H 3 rad ( 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.