Blowup alternative for Gross-Pitaevskii hierarchies

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.