On the Cauchy problem for focusing and defocusing Gross-Pitaevskiihierarchies

We consider thedynamical Gross-Pitaevskii (GP) hierarchy on ᵈ, d≥1,for cubic, quintic, focusing and defocusing interactions.For both the focusing and defocusing case, and any d≥1,we prove localexistence and uniqueness of solutions in certainSobolev type spaces _ξ^α 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 α > 1/2 if d=1 α > d2-1/2(p-1) if d≥2 and (d,p)≠(3,2) α ≥ 1 if (d,p)=(3,2)where $p=2$ for the cubic, and $p=4$ for the quintic GP hierarchy;the parameter ξ>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 L² criticality,both in the focusing and defocusing context.All of these results hold without the assumption of factorized initial conditions.