08-239 Thomas Chen and Natasa Pavlovic
On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies (374K, pdf) Dec 18, 08
Abstract , Paper (src), View paper (auto. generated pdf), Index of related papers

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.

Files: 08-239.src( 08-239.comments , 08-239.keywords , boseNLSqc.pdf.mm )