On Blow-up Solutions to the 3d Cubic Nonlinear Schrödinger Equation

For the 3d cubic nonlinear Schrödinger (NLS) equation, which has critical (scaling) norms L and Ḣ, 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 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 ∼ (T − t), 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 ∼ (T − t), but focus towards this sphere at a faster rate ∼ (T − t). 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.