Energy conservation and blowup of solutions for focusing Gross–Pitaevskii hierarchies
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 5, p. 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.
@article{AIHPC_2010__27_5_1271_0,
author = {Chen, Thomas and Pavlovi\'c, Nata\v sa and Tzirakis, Nikolaos},
title = {Energy conservation and blowup of solutions for focusing Gross--Pitaevskii hierarchies},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {27},
number = {5},
year = {2010},
pages = {1271-1290},
doi = {10.1016/j.anihpc.2010.06.003},
zbl = {1200.35253},
mrnumber = {2683760},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2010__27_5_1271_0}
}

Chen, Thomas; Pavlović, Nataša; Tzirakis, Nikolaos. Energy conservation and blowup of solutions for focusing Gross–Pitaevskii hierarchies. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 5, pp. 1271-1290. doi : 10.1016/j.anihpc.2010.06.003. http://www.numdam.org/item/AIHPC_2010__27_5_1271_0/

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