A new proof of existence of solutions for focusing and defocusing Gross-Pitaevskii hierarchies
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- by Thomas Chen and Nataša Pavlović PDF
- Proc. Amer. Math. Soc. 141 (2013), 279-293 Request permission
Abstract:
We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in $d\geq 1$ dimensions for focusing and defocusing interactions. We present a new proof of existence of solutions that does not require the a priori bound on the spacetime norm, which was introduced in the work of Klainerman and Machedon and used in our earlier work.References
- Riccardo Adami, François Golse, and Alessandro Teta, Rigorous derivation of the cubic NLS in dimension one, J. Stat. Phys. 127 (2007), no. 6, 1193–1220. MR 2331036, DOI 10.1007/s10955-006-9271-z
- M. Aizenman, E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason, Bose-Einstein Quantum Phase Transition in an Optical Lattice Model, Phys. Rev. A 70, 023612 (2004).
- I. Anapolitanos, I.M. Sigal, The Hartree-von Neumann limit of many body dynamics, Preprint http://arxiv.org/abs/0904.4514.
- Thierry Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 2002047, DOI 10.1090/cln/010
- Thomas Chen and Nataša Pavlović, The quintic NLS as the mean field limit of a boson gas with three-body interactions, J. Funct. Anal. 260 (2011), no. 4, 959–997. MR 2747009, DOI 10.1016/j.jfa.2010.11.003
- Thomas Chen and Nataša Pavlović, On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies, Discrete Contin. Dyn. Syst. 27 (2010), no. 2, 715–739. MR 2600687, DOI 10.3934/dcds.2010.27.715
- Thomas Chen, Nataša Pavlović, and Nikolaos Tzirakis, Energy conservation and blowup of solutions for focusing Gross-Pitaevskii hierarchies, Ann. Inst. H. Poincaré C Anal. Non Linéaire 27 (2010), no. 5, 1271–1290. MR 2683760, DOI 10.1016/j.anihpc.2010.06.003
- Zeqian Chen and Chuangye Liu, On the Cauchy problem for Gross-Pitaevskii hierarchies, J. Math. Phys. 52 (2011), no. 3, 032103, 13. MR 2814691, DOI 10.1063/1.3567168
- Alexander Elgart, László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons, Arch. Ration. Mech. Anal. 179 (2006), no. 2, 265–283. MR 2209131, DOI 10.1007/s00205-005-0388-z
- László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate, Comm. Pure Appl. Math. 59 (2006), no. 12, 1659–1741. MR 2257859, DOI 10.1002/cpa.20123
- László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math. 167 (2007), no. 3, 515–614. MR 2276262, DOI 10.1007/s00222-006-0022-1
- László Erdős and Horng-Tzer Yau, Derivation of the nonlinear Schrödinger equation from a many body Coulomb system, Adv. Theor. Math. Phys. 5 (2001), no. 6, 1169–1205. MR 1926667, DOI 10.4310/ATMP.2001.v5.n6.a6
- Jürg Fröhlich, Sandro Graffi, and Simon Schwarz, Mean-field- and classical limit of many-body Schrödinger dynamics for bosons, Comm. Math. Phys. 271 (2007), no. 3, 681–697. MR 2291792, DOI 10.1007/s00220-007-0207-5
- J. Fröhlich, A. Knowles, and A. Pizzo, Atomism and quantization, J. Phys. A 40 (2007), no. 12, 3033–3045. MR 2313859, DOI 10.1088/1751-8113/40/12/S09
- Jürg Fröhlich, Antti Knowles, and Simon Schwarz, On the mean-field limit of bosons with Coulomb two-body interaction, Comm. Math. Phys. 288 (2009), no. 3, 1023–1059. MR 2504864, DOI 10.1007/s00220-009-0754-z
- Manoussos G. Grillakis, Matei Machedon, and Dionisios Margetis, Second-order corrections to mean field evolution of weakly interacting bosons. I, Comm. Math. Phys. 294 (2010), no. 1, 273–301. MR 2575484, DOI 10.1007/s00220-009-0933-y
- Manoussos G. Grillakis and Dionisios Margetis, A priori estimates for many-body Hamiltonian evolution of interacting boson system, J. Hyperbolic Differ. Equ. 5 (2008), no. 4, 857–883. MR 2475483, DOI 10.1142/S0219891608001726
- Klaus Hepp, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys. 35 (1974), 265–277. MR 332046
- Sergiu Klainerman and Matei Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Comm. Math. Phys. 279 (2008), no. 1, 169–185. MR 2377632, DOI 10.1007/s00220-008-0426-4
- Kay Kirkpatrick, Benjamin Schlein, and Gigliola Staffilani, Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics, Amer. J. Math. 133 (2011), no. 1, 91–130. MR 2752936, DOI 10.1353/ajm.2011.0004
- E.H. Lieb, R. Seiringer, Proof of Bose-Einstein condensation for dilute trapped gases, Phys. Rev. Lett. 88, 170409 (2002).
- Elliott H. Lieb, Robert Seiringer, Jan Philip Solovej, and Jakob Yngvason, The mathematics of the Bose gas and its condensation, Oberwolfach Seminars, vol. 34, Birkhäuser Verlag, Basel, 2005. MR 2143817
- Elliott H. Lieb, Robert Seiringer, and Jakob Yngvason, A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas, Comm. Math. Phys. 224 (2001), no. 1, 17–31. Dedicated to Joel L. Lebowitz. MR 1868990, DOI 10.1007/s002200100533
- Igor Rodnianski and Benjamin Schlein, Quantum fluctuations and rate of convergence towards mean field dynamics, Comm. Math. Phys. 291 (2009), no. 1, 31–61. MR 2530155, DOI 10.1007/s00220-009-0867-4
- 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.
- Herbert Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. Modern Phys. 52 (1980), no. 3, 569–615. MR 578142, DOI 10.1103/RevModPhys.52.569
- Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR 2233925, DOI 10.1090/cbms/106
Additional Information
- Thomas Chen
- Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
- Email: tc@math.utexas.edu
- Nataša Pavlović
- Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
- MR Author ID: 697878
- Email: natasa@math.utexas.edu
- Received by editor(s): December 22, 2010
- Received by editor(s) in revised form: June 17, 2011
- Published electronically: May 18, 2012
- Communicated by: Hart F. Smith
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 279-293
- MSC (2010): Primary 35Q55
- DOI: https://doi.org/10.1090/S0002-9939-2012-11308-5
- MathSciNet review: 2988730