On Blow-up Solutions to the 3D Cubic Nonlinear Schrodinger Equation

For the 3D cubic nonlinear Schrodinger (NLS) equation, which has critical (scaling) norms L-3 and 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 L3 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 similar to(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," that is, those that concentrate on a sphere of radius similar to(T-t)(1/3), but focus toward this sphere at a faster rate similar to(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.