Negative energy blowup results for the focusing Hartree hierarchy via identities of virial and localized virial type

TL;DR

This paper proves finite-time blowup for negative energy solutions of the focusing Hartree hierarchy using virial identities, localized virial estimates, and quantum de Finetti theorem, extending blowup results to cases without finite variance.

Contribution

It introduces virial and localized virial identities for the Hartree hierarchy and demonstrates negative energy blowup without finite variance assumptions.

Findings

Negative energy solutions blow up in finite time.

Virial identities are established for the Hartree hierarchy.

Localized virial estimates are effective without finite variance.

Abstract

We establish virial and localized virial identities for solutions to the Hartree hierarchy, an infinite system of partial differential equations which arises in mathematical modeling of many body quantum systems. As an application, we use arguments originally developed in the study of the nonlinear Schr\"odinger equation (see work of Zakharov, Glassey, and Ogawa--Tsutsumi) to show that certain classes of negative energy solutions must blow up in finite time. The most delicate case of this analysis is the proof of negative energy blowup without the assumption of finite variance; in this case, we make use of the localized virial estimates, combined with the quantum de Finetti theorem of Hudson and Moody and several algebraic identities adapted to our particular setting. Application of a carefully chosen truncation lemma then allows for the additional terms produced in the localization argument to be controlled.