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

Thomas Chen; Nataša Pavlović

paper

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

arXiv:0812.3463

Abstract

We consider the dynamical Gross-Pitaevskii (GP) hierarchy on $R^{d}$ , $d \geq 1$ , for cubic, quintic, focusing and defocusing interactions. For both the focusing and defocusing case, and any $d \geq 1$ , we prove local existence and uniqueness of solutions in certain Sobolev type spaces $\cH_ξ^α$ of sequences of marginal density matrices. The regularity is accounted for by $α>\frac12& if $d=1 $,$ α>\frac d2-\frac{1}{2(p-1)} $i f$ d\geq2 $an d$ (d,p)\neq(3,2) $, an d$ α\geq1 $i f$ (d,p)=(3,2) $, w h er e$ p=2 $f or t h ec u bi c, an d$ p=4 $f or t h e q u in t i c GP hi er a r c h y; t h e p a r am e t er$ ξ>0 $i s a r bi t r a r y an dd e t er min es t h ee n er g y sc a l eo f t h e p r o b l e m . T hi sr es u l t in c l u d es t h e p r oo f o f ana p r i or i s p a ce t im e b o u n d co nj ec t u r e d b y K l ain er manan d M a c h e d o n f or t h ec u bi c GP hi er a r c h y in$ d=3 $. I n t h e d e f oc u s in g c a se, w e p r o v e t h ee x i s t e n ce an d u ni q u e n esso f so l u t i o n s g l o ba l l y in t im e f or t h ec u bi c GP hi er a r c h y f or$ 1\leq d\leq3 $, an d o f t h e q u in t i c GP hi er a r c h y f or$ 1\leq d\leq 2 $, inana pp r o p r ia t es p a ceo f S o b o l e v t y p e, an d u n d er t h e a ss u m pt i o n o f ana p r i or i e n er g y b o u n d . F or t h e f oc u s in g GP hi er a r c hi es, w e p r o v e l o w er b o u n d so n t h e b l o w u p r a t e . A l so p se u d oco n f or ma l in v a r ian ce i ses t ab l i s h e d in t h ec a sescor r es p o n d in g t o$ L^2$ criticality, both in the focusing and defocusing context. All of these results hold without the assumption of factorized initial conditions.

AMS Latex, 28 pages

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

Abstract

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