Blowup alternative for Gross-Pitaevskii hierarchies

TL;DR

This paper establishes finite-time blowup conditions for solutions to the focusing Gross-Pitaevskii hierarchy in higher dimensions, providing lower bounds on blowup rates without requiring factorized initial states.

Contribution

It proves the blowup alternative and lower bounds on blowup rates for the hierarchy without initial factorization assumptions.

Findings

Solutions blow up in finite time if initial energy per k particles is negative for n ≥ 3.

Results hold without factorized initial conditions.

Provides lower bounds on blowup rates.

Abstract

In this paper, we prove the blowup alternative for Gross-Pitaevskii hierarchies on $\mathbb{R}^n$ and give the associated lower bounds on the blowup rate. In particular, we prove that any solution of density operators to the focusing Gross-Pitaevskii hierarchy blow up in finite time for $n \ge 3$ if the energy per some $k$ particles in the initial condition is negative. All of these results hold without the assumption of factorized conditions for initial values as well as the admissible ones. Our analysis is based on use of a quasi-Banach space of sequences of marginal density matrices.