Mathematics – Analysis of PDEs
Scientific paper
2007-03-08
Mathematics
Analysis of PDEs
Scientific paper
For the 3d cubic nonlinear Schr\"odinger (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.
Holmer Justin
Roudenko Svetlana
No associations
LandOfFree
On blow-up solutions to the 3D cubic nonlinear Schroedinger equation does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On blow-up solutions to the 3D cubic nonlinear Schroedinger equation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On blow-up solutions to the 3D cubic nonlinear Schroedinger equation will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-642802