  The aim of this paper is to study, in dimensions 2 and 3, the pure-power
non-linear Schr\"odinger equation with an external uniform magnetic field
included. In particular, we derive a general criteria on the initial data and
the power of the non-linearity so that the corresponding solution blows up in
finite time, and we show that the time for blow up to occur decreases as the
strength of the magnetic field increases. In addition, we also discuss some
observations about Strichartz estimates in 2 dimensions for the Mehler kernel,
as well as similar blow-up results for the non-linear Pauli equation.
