聚焦Gross-Pitaevskii方程组解的能量守恒和爆破
Energy conservation and blowup of solutions for focusing Gross-Pitaevskii hierarchies
摘要：
我们认为解决方案的重点三次和五次Gross-Pitaevskii（GP）的层次。我们确定了一个可观察到的对应于每个粒子的平均能量，我们证明了它是一个守恒量。证明了当初始条件下每个粒子的能量为负时，在$L^2$ -临界或$L^2$ -超临界水平上聚焦GP族的所有解在有限时间内爆破.我们的结果不假设任何分解的初始数据。
英文摘要：
We consider solutions of the focusing cubic and quintic Gross-Pitaevskii (GP) hierarchies. We identify an observable corresponding to the average energy per particle, and we prove that it is a conserved quantity. We prove that all solutions to the focusing GP hierarchy at the $L^2$-critical or $L^2$-supercritical level blow up in finite time if the energy per particle in the initial condition is negative. Our results do not assume any factorization of the initial data.
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