Thomas Chen and Natasa Pavlovic
On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies
(374K, pdf)
ABSTRACT. 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 wellposedness of the Cauchy problem in weighted Sobolev spaces $\cH_\xi^\alpha$ of sequences of marginal density matrices, for $\alpha>\frac12& if $d=1$, $\alpha>\frac d2-\frac{1}{2(p-1)} $ if $d\geq2$ and $(d,p)\neq(3,2)$, and $\alpha\geq1$ 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. This result includes the proof of an a priori spacetime bound conjectured by Klainerman and Machedon for the cubic GP hierarchy in $d=3$. In the defocusing case, we prove global wellposedness in $\cH_\xi^1$ of the cubic GP hierarchy for $1\leq d\leq3$, and of the quintic GP hierarchy for $1\leq d\leq 2$. For the focusing GP hierarchies, we prove lower bounds on the blowup rate, and pseudoconformal invariance in the cases corresponding to $L^2$ criticality. All of these results hold without the assumption of factorized initial conditions.