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

Thomas Chen; Nataša Pavlović; Nikolaos Tzirakis

doi:10.1016/j.anihpc.2010.06.003

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

Thomas Chen, Nataša Pavlović, Nikolaos Tzirakis

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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 L²-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.

Original language	English (US)
Pages (from-to)	1271-1290
Number of pages	20
Journal	Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume	27
Issue number	5
DOIs	https://doi.org/10.1016/j.anihpc.2010.06.003
State	Published - 2010

ASJC Scopus subject areas

Analysis
Mathematical Physics
Applied Mathematics

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10.1016/j.anihpc.2010.06.003

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title = "Energy conservation and blowup of solutions for focusing Gross-Pitaevskii hierarchies",

abstract = "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 L2-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.",

author = "Thomas Chen and Nata{\v s}a Pavlovi{\'c} and Nikolaos Tzirakis",

note = "Funding Information: T.C. thanks I. Rodnianski for inspiring discussions. We thank A. Bulut for proofreading an earlier version of this manuscript. The work of T.C. was supported by NSF grant DMS-0704031/DMS-0940145. The work of N.P. was supported by NSF grant number DMS-0758247 and an Alfred P. Sloan Research Fellowship. The work of N.T. was supported by NSF grant DMS-0901222.",

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language = "English (US)",

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T1 - Energy conservation and blowup of solutions for focusing Gross-Pitaevskii hierarchies

AU - Chen, Thomas

AU - Pavlović, Nataša

AU - Tzirakis, Nikolaos

N1 - Funding Information: T.C. thanks I. Rodnianski for inspiring discussions. We thank A. Bulut for proofreading an earlier version of this manuscript. The work of T.C. was supported by NSF grant DMS-0704031/DMS-0940145. The work of N.P. was supported by NSF grant number DMS-0758247 and an Alfred P. Sloan Research Fellowship. The work of N.T. was supported by NSF grant DMS-0901222.

PY - 2010

Y1 - 2010

N2 - 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 L2-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.

AB - 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 L2-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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ER -

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

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