Non-linear Schrödinger Equation in a uniform magnetic field

Abstract: 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.