On blow-up solutions to the 3D cubic nonlinear Schroedinger equation

Justin Holmer; Svetlana Roudenko

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arXiv · math/0703233

On blow-up solutions to the 3D cubic nonlinear Schroedinger equation

Justin Holmer·Svetlana Roudenko

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Abstract

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.

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Justin Holmer, Svetlana Roudenko. 2007-04-04. On blow-up solutions to the 3D cubic nonlinear Schroedinger equation. https://arxiv.org/abs/math/0703233

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