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On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies

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  • We consider the dynamical 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 local existence and uniqueness of solutions in certain Sobolev type spaces $\H_\xi^\alpha$ of sequences of marginal density matrices which satisfy the space-time bound conjectured by 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.

    Mathematics Subject Classification: Primary: 35Q55, 81V70.

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