Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-12T16:14:20.836Z Has data issue: false hasContentIssue false
  • English
  • Français

Recent Results on the Cauchy Problem for Focusing andDefocusing Gross-Pitaevskii Hierarchies

Published online by Cambridge University Press:  12 May 2010

Thomas Chen  and
Nataša Pavlović
Get access

Abstract

In this paper, we review some of our recent results in the study of the dynamics ofinteracting Bose gases in the Gross-Pitaevskii (GP) limit. Our investigations focus on thewell-posedness of the associated Cauchy problem for the infinite particle system describedby the GP hierarchy.

Keywords

Bose gasGross-Pitaevskii limitBBGKY hierarchynonlinear Schrödinger equationsmean field limit
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 4: Spectral problems. Issue dedicated to the memory of M. Birman , 2010 , pp. 54 - 72
Copyright
© EDP Sciences, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adami, R., Golse, G., Teta, A.. Rigorous derivation of the cubic NLS in dimension one . J. Stat. Phys., 127 (2007), No. 6, 1194–1220.CrossRefGoogle Scholar
Aizenman, M., Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J.. Bose-Einstein Quantum Phase Transition in an Optical Lattice Model . Phys. Rev. A, 70 (2004), 023612.CrossRefGoogle Scholar
I. Anapolitanos, I.M. Sigal. The Hartree-von Neumann limit of many body dynamics. Preprint http://arxiv.org/abs/0904.4514.
Bach, V., Chen, T., Fröhlich, J. andSigal, I.M.. Smooth Feshbach map and operator-theoretic renormalization group methods . J. Funct. Anal., 203 (2003), No. 1, 44-92. CrossRefGoogle Scholar
T. Cazenave. Semilinear Schrödinger equations. Courant lecture notes, 10 (2003), Amer. Math. Soc..
T. Chen, N. Pavlović. The quintic NLS as the mean field limit of a Boson gas with three-body interactions. J. Functional Analysis, conditionally accepted. Preprint http://arxiv.org/abs/0812.2740.
Chen, T., Pavlović, N.. On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies . Discr. Contin. Dyn. Syst., 27 (2010), No. 2, 715 - 739. Google Scholar
T. Chen, N. Pavlović. A short proof of local wellposedness for focusing and defocusing Gross-Pitaevskii hierarchies. Preprint http://arxiv.org/abs/0906.3277.
T. Chen, N. Pavlović. Higher order energy conservation, Gagliardo-Nirenberg-Sobolev inequalities, and global well-posedness for Gross-Pitaevskii hierarchies. Preprint http://arxiv.org/abs/0906.2984.
T. Chen, N. Pavlović, N. Tzirakis. Energy conservation and blowup of solutions for focusing GP hierarchies. Preprint http://arXiv.org/abs/0905.2704.
Elgart, A., Erdös, L., Schlein, B., Yau, H.-T.. Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons . Arch. Rat. Mech. Anal., 179 (2006), No. 2, 265–283. CrossRefGoogle Scholar
Erdös, L., Schlein, B., Yau, H.-T.. Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate . Comm. Pure Appl. Math., 59 (2006), No. 12, 1659–1741. CrossRefGoogle Scholar
Erdös, L., Schlein, B., Yau, H.-T.. Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems . Invent. Math., 167 (2007), 515–614. CrossRefGoogle Scholar
Erdös, L., Yau, H.-T.. Derivation of the nonlinear Schrödinger equation from a many body Coulomb system . Adv. Theor. Math. Phys., 5 (2001), No. 6, 1169–1205. CrossRefGoogle Scholar
Fröhlich, J., Graffi, S., Schwarz, S.. Mean-field- and classical limit of many-body Schrödinger dynamics for bosons . Comm. Math. Phys., 271 (2007), no. 3, 681–697. CrossRefGoogle Scholar
Fröhlich, J., Knowles, A., Pizzo, A.. Atomism and quantization . J. Phys. A, 40 (2007), No. 12, 3033–3045. CrossRefGoogle Scholar
J. Fröhlich, A. Knowles, S. Schwarz. On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction. Preprint arXiv:0805.4299.
M. Grillakis, M. Machedon, A. Margetis. Second-order corrections to mean field evolution for weakly interacting Bosons I. Preprint http://arxiv.org/abs/0904.0158.
Grillakis, M., Margetis, A.. A priori estimates for many-body Hamiltonian evolution of interacting boson system . J. Hyperbolic Differ. Equ., 5 (2008), No. 4, 857–883. CrossRefGoogle Scholar
Hepp, K.. The classical limit for quantum mechanical correlation functions . Comm. Math. Phys., 35 (1974), 265–277.CrossRefGoogle Scholar
Klainerman, S., Machedon, M.. On the uniqueness of solutions to the Gross-Pitaevskii hierarchy . Commun. Math. Phys., 279 (2008), No. 1, 169–185. CrossRefGoogle Scholar
K. Kirkpatrick, B. Schlein, G. Staffilani. Derivation of the two dimensional nonlinear Schrödinger equation from many body quantum dynamics. Preprint arXiv:0808.0505.
Lieb, E.H., Seiringer, R.. Proof of Bose-Einstein condensation for dilute trapped gases . Phys. Rev. Lett., 88 (2002), 170409. CrossRefGoogle ScholarPubMed
E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason. The mathematics of the Bose gas and its condensation. Birkhäuser (2005).
Lieb, E.H., Seiringer, R., Yngvason, J.. A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas . Commun. Math. Phys., 224 (2001) No. 1, 17–31. CrossRefGoogle Scholar
Rodnianski, I., Schlein, B.. Quantum fluctuations and rate of convergence towards mean field dynamics . Comm. Math. Phys., 29 (2009), No. 1, 31–611. CrossRefGoogle Scholar
B. Schlein. Derivation of Effective Evolution Equations from Microscopic Quantum Dynamics. Lecture notes for the minicourse held at the 2008 CMI Summer School in Zurich.
Spohn, H.. Kinetic Equations from Hamiltonian Dynamics . Rev. Mod. Phys. 52 (1980), No. 3, 569–615. CrossRefGoogle Scholar
T. Tao.Nonlinear dispersive equations. Local and global analysis. CBMS 106 (2006), AMS.