We prove that if
is a log-log blow-up solution, of the type studied by Merle and Raphaël, to the
critical focusing
NLS equation
with initial data
in the cases
,
then
remains
bounded in
away from the blow-up point. This is obtained without assuming that the initial data
has any regularity
beyond
. As an
application of the
result, we construct an open subset of initial data in the radial energy space
with
corresponding solutions that blow up on a sphere at positive radius for the 3D quintic
(-critical) focusing
NLS equation
.
This improves the results of Raphaël and Szeftel [2009], where an open subset in
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.