Well-posedness for Gross-pitaevskii Hierarchies

We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchy in d dimensions, for focusing and defocusing interactions. We introduce new higher order conserved energy functionals that allow us to prove global existence and uniqueness of solutions for defocusing GP hierarchies, with arbitrary initial data in the energy space. Moreover, we prove generalizations of the Sobolev and Gagliardo-Nirenberg inequalities for density matrices, which we apply to establish global existence and uniqueness of solutions for focusing and defocusing GP hierarchies on the L-subcritical level.