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

TL;DR

This paper studies focusing Gross-Pitaevskii hierarchies, showing that solutions with negative initial energy blow up in finite time without requiring initial data factorization.

Contribution

It introduces an energy observable for focusing GP hierarchies and proves finite-time blowup for solutions with negative initial energy at critical or supercritical levels.

Findings

Energy per particle is conserved.

Negative initial energy leads to finite-time blowup.

Results hold without initial data factorization.

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 $L^2$-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.