  For the 3d cubic nonlinear Schr\"odinger (NLS) equation, which has critical
(scaling) norms $L^3$ and $\dot H^{1/2}$, we first prove a result establishing
sufficient conditions for global existence and sufficient conditions for
finite-time blow-up. For the rest of the paper, we focus on the study of
finite-time radial blow-up solutions, and prove a result on the concentration
of the $L^3$ norm at the origin. Two disparate possibilities emerge, one which
coincides with solutions typically observed in numerical experiments that
consist of a specific bump profile with maximum at the origin and focus toward
the origin at rate $\sim(T-t)^{1/2}$, where $T>0$ is the blow-up time. For the
other possibility, we propose the existence of ``contracting sphere blow-up
solutions'', i.e. those that concentrate on a sphere of radius $\sim
(T-t)^{1/3}$, but focus towards this sphere at a faster rate $\sim
(T-t)^{2/3}$. These conjectured solutions are analyzed through heuristic
arguments and shown (at this level of precision) to be consistent with all
conservation laws of the equation.

Analysis of PDEs