On the Cauchy problem for focusing and defocusing Gross-Pitaevskiihierarchies

Abstract

We consider thedynamical Gross-Pitaevskii (GP) hierarchy on $\R^d$, $d\geq1$,for cubic, quintic, focusing and defocusing interactions.For both the focusing and defocusing case, and any $d\geq1$,we prove localexistence and uniqueness of solutions in certainSobolev type spaces $\H_\xi^\alpha$ of sequences of marginaldensity matrices which satisfy the space-time bound conjecturedby Klainerman and Machedon for the cubic GP hierarchy in $d=3$.The regularity is accounted for by $ \alpha $ > 1/2 if d=1 $ \alpha > \frac d2-\frac{1}{2(p-1)} if d\geq2 and (d,p)\neq(3,2) $ $ \alpha \geq 1 if (d,p)=(3,2) $where $p=2$ for the cubic, and $p=4$ for the quintic GP hierarchy;the parameter $\xi>0$ is arbitrary and determines the energy scale of the problem.For focusing GP hierarchies, we prove lower bounds on the blowup rate.Moreover, pseudoconformal invariance is established in the cases corresponding to $L^2$ criticality,both in the focusing and defocusing context.All of these results hold without the assumption of factorized initial conditions.