Higher order energy conservation and global wellposedness of solutions for Gross-Pitaevskii hierarchies

We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in $d$ dimensions, for focusing and defocusing interactions. We introduce new higher order energy functionals and prove that they are conserved for solutions of energy subcritical defocusing, and $L^2$ subcritical (de)focusing GP hierarchies, in spaces also used by Erd\"os, Schlein and Yau in \cite{esy1,esy2}. By use of this tool, we prove a priori $H^1$ bounds for positive semidefinite solutions in those spaces. Moreover, we obtain global well-posedness results for positive semidefinite solutions in the spaces studied in the works of Klainerman and Machedon, \cite{klma}, and in \cite{chpa2}. As part of our analysis, we prove generalizations of Sobolev and Gagliardo-Nirenberg inequalities for density matrices.