Standing Ring Blowup Solutions for Cubic NLS

摘要

We prove there exist solutions to the focusing cubic nonlinear Schr\"odingerequation in three dimensions that blowup on a circle, in the sense of L^2concentration on a ring, bounded H^1 norm outside any surrounding toroid, andgrowth of the global H^1 norm with the log-log rate. Analogous behaviour occursin any higher dimension. That is, there exists data for which the correspondingevolution by cubic NLS explodes on a set of co-dimension two. To simplify theexposition, the proof is presented in dimension three, with remarks to indicatethe adaptations in higher dimension.