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 -critical or -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{s}a and Tzirakis, Nikolaos}, title = {Energy conservation and blowup of solutions for focusing {Gross{\textendash}Pitaevskii} hierarchies}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {1271--1290}, publisher = {Elsevier}, volume = {27}, number = {5}, year = {2010}, doi = {10.1016/j.anihpc.2010.06.003}, mrnumber = {2683760}, zbl = {1200.35253}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2010.06.003/} }
TY - JOUR AU - Chen, Thomas AU - Pavlović, Nataša AU - Tzirakis, Nikolaos TI - Energy conservation and blowup of solutions for focusing Gross–Pitaevskii hierarchies JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 1271 EP - 1290 VL - 27 IS - 5 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2010.06.003/ DO - 10.1016/j.anihpc.2010.06.003 LA - en ID - AIHPC_2010__27_5_1271_0 ER -
%0 Journal Article %A Chen, Thomas %A Pavlović, Nataša %A Tzirakis, Nikolaos %T Energy conservation and blowup of solutions for focusing Gross–Pitaevskii hierarchies %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 1271-1290 %V 27 %N 5 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2010.06.003/ %R 10.1016/j.anihpc.2010.06.003 %G en %F 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://archive.numdam.org/articles/10.1016/j.anihpc.2010.06.003/
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