Higher Order Energy Conservation and Global Well-Posedness of Solutions for Gross-Pitaevskii Hierarchies

Abstract

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 Erdös, Schlein and Yau. 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. As part of our analysis, we prove generalizations of Sobolev and Gagliardo-Nirenberg inequalities for density matrices.