DOI: 10.3724/SP.J.1160.2013.00291

Acta Analysis Functionalis Applicata (应用泛函分析学报) 2013/15:4 PP.291-305

Local Well-posedness for Gross-Pitaevskii Hierarchies

We consider the Cauchy problem for the Gross-Pitaevskii infinite linear hierarchy of equations on R~n.By introducing a(F)-norm in certain Sobolev type spaces of sequences of marginal density matrices,we establish local existence,uniqueness and stability of solutions.Explicit spacetime type estimates for the solutions are obtained as well.In particular,this(F)-norm is compatible with the usual Sobolev space norm whenever the initial data is factorized.

Key words:Gross-Pitaevskii hierarchy,Cauchy problem,local well-posedness

ReleaseDate:2015-04-21 14:22:06

1 Dalfovo F,Giorgini S,Pitaevskii L P. Theory of Bose-Einstein condensation in trapped gases[J].Re v Mod Phys,1999.463-512.

2 Erdös L,Schlein B,YauH T. Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems[J].Inventiones Mathematicae,2007.515-614.

3 Erdös L,Schlein B,Yau H T. Rigorous derivation of the Gross-Pitaevskii equation[J].Physical Review Letters,2007.040404.

4 Erdös L,Schlein B,Yau H T. Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate[J].ANNALS OF MATHEMATICS,2010.291-370.

5 Chen T,Pavlovi(c) N. On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies[J].DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,2010,(02):715-739.

6 Chen Z,Liu C. On the Cauchy problem for Gross-Pitaevskii hierarchies[J].Journal of Mathematical Physics,2011.032103.

7 Cazenave T. Semilinear Schrödinger equations[M].Providence:American Mathematical Society,2003.

8 Köthe G. Topological vector spaces I[M].New York:Springer-Verlag,1983.

9 Tao T. Nonlinear Dispersive Equations:Local and Global Ananlysis[M].Providence:American Mathematical Society,2006.

10 Klainerman S,Machedon M. On the uniqueness of solutions to the Gross-Pitaevskii hierarchy[J].COMMUNICATIONS IN MATHEMATICAL PHYSICS,2008.169-185.

11 Chen T,Pavlović N. The quintic NLS as the mean field limit of a Boson gas with three-body interactions[J].Journal of Functional Analysis,2011,(04):959-997.