On blow-up solutions to the 3D cubic nonlinear Schroedinger equation
| dc.creator | Holmer, Justin | |
| dc.creator | Roudenko, Svetlana | |
| dc.date | 2007-03-08 | |
| dc.date | 2007-04-04 | |
| dc.date.accessioned | 2026-07-07T07:55:00Z | |
| dc.date.available | 2026-07-07T07:55:00Z | |
| dc.description | For the 3d cubic nonlinear Schrödinger (NLS) equation, which has critical (scaling) norms $L^3$ and $\dot H^{1/2}$, 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 $L^3$ 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 $\sim(T-t)^{1/2}$, where $T>0$ is the blow-up time. For the other possibility, we propose the existence of ``contracting sphere blow-up solutions'', i.e. those that concentrate on a sphere of radius $\sim (T-t)^{1/3}$, but focus towards this sphere at a faster rate $\sim (T-t)^{2/3}$. 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. | |
| dc.identifier | https://arxiv.org/abs/math/0703233 | |
| dc.identifier | http://arxiv.org/abs/math/0703233 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/126828 | |
| dc.subject | Analysis of PDEs | |
| dc.title | On blow-up solutions to the 3D cubic nonlinear Schroedinger equation | |
| dc.type | text |