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

Aynur Bulut

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 Schrödinger 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.

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

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 Schrödinger 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.

Paper Structure

This paper contains 7 sections, 11 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A momentum identity and mass conservation
  4. Conservation of Energy
  5. Virial and localized virial identities
  6. Proofs of the main theorems: negative energy blow-up solutions
  7. Localized virial identities adapted to scaling

Key Result

Theorem \oldthetheorem

Fix $d\geq 2$, $\mu=-1$ and suppose that $V\in\mathcal{S}(\mathbb{R}^d;\mathbb{R})$ is a bounded even function such that Let $(\gamma^{(k)})_{k\geq 1}$ be an (A)--(D)-admissible solution to (label_1) defined on an interval $I\subset\mathbb{R}$ with Then $I$ is bounded.

Theorems & Definitions (22)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem: Conservation of mass for (\ref{['label_1']})
  • proof
  • Proposition \oldthetheorem: Conservation of energy for (\ref{['label_1']})
  • proof
  • Remark \oldthetheorem
  • ...and 12 more