Energy conservation and blowup of solutions for focusing Gross–Pitaevskii hierarchies

Chen, Thomas; Pavlović, Nataša; Tzirakis, Nikolaos

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 5, pp. 1271-1290.

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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