Physics – Mathematical Physics
Scientific paper
2008-12-18
Discr. Contin. Dyn. Syst., 27 (2), 715 - 739, 2010
Physics
Mathematical Physics
AMS Latex, 28 pages
Scientific paper
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 $\cH_\xi^\alpha$ of sequences of marginal density matrices. The regularity is accounted for by $\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 the existence and uniqueness of solutions globally in time for the cubic GP hierarchy for $1\leq d\leq3$, and of the quintic GP hierarchy for $1\leq d\leq 2$, in an appropriate space of Sobolev type, and under the assumption of an a priori energy bound. For the focusing GP hierarchies, we prove lower bounds on the blowup rate. Also 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.
Chen Thomas
Pavlović Nataša
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