Energy conservation and blowup of solutions for focusing Gross–Pitaevskii hierarchies
Chen, Thomas ; Pavlović, Nataša ; Tzirakis, Nikolaos
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010), p. 1271-1290 / Harvested from Numdam
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We consider solutions of the focusing cubic and quintic Gross–Pitaevskii (GP) hierarchies. We identify an observable corresponding to the average energy per particle, and we prove that it is a conserved quantity. We prove that all solutions to the focusing GP hierarchy at the L 2 -critical or L 2 -supercritical level blow up in finite time if the energy per particle in the initial condition is negative. Our results do not assume any factorization of the initial data.

@article{AIHPC_2010__27_5_1271_0,
     author = {Chen, Thomas and Pavlovi\'c, Nata\v sa and Tzirakis, Nikolaos},
     title = {Energy conservation and blowup of solutions for focusing Gross--Pitaevskii hierarchies},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {27},
     year = {2010},
     pages = {1271-1290},
     doi = {10.1016/j.anihpc.2010.06.003},
     mrnumber = {2683760},
     zbl = {1200.35253},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2010__27_5_1271_0}
}

Chen, Thomas; Pavlović, Nataša; Tzirakis, Nikolaos. Energy conservation and blowup of solutions for focusing Gross–Pitaevskii hierarchies. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) pp. 1271-1290. doi : 10.1016/j.anihpc.2010.06.003. http://gdmltest.u-ga.fr/item/AIHPC_2010__27_5_1271_0/

[1] R. Adami, G. Golse, A. Teta, Rigorous derivation of the cubic NLS in dimension one, J. Stat. Phys. 127 no. 6 (2007), 1194-1220 | MR 2331036 | Zbl 1118.81021

[2] 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 (2004), 023612 | Zbl 1170.82325

[3] I. Anapolitanos, I.M. Sigal, The Hartree–von Neumann limit of many body dynamics, http://arxiv.org/abs/0904.4514 | Zbl 1339.81020

[4] V. Bach, T. Chen, J. Fröhlich, I.M. Sigal, Smooth Feshbach map and operator-theoretic renormalization group methods, J. Funct. Anal. 203 no. 1 (2003), 44-92 | MR 1996868 | Zbl 1060.47028

[5] T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes vol. 10, Amer. Math. Soc. (2003) | MR 2002047 | Zbl 1055.35003

[6] T. Chen, N. Pavlović, The quintic NLS as the mean field limit of a Boson gas with three-body interactions, http://arxiv.org/abs/0812.2740 | MR 2747009 | Zbl 1213.35368

[7] T. Chen, N. Pavlović, On the Cauchy problem for focusing and defocusing Gross–Pitaevskii hierarchies, Discrete Contin. Dyn. Syst. Ser. A 27 no. 2 (2010), 715-739 | MR 2600687 | Zbl 1190.35207

[8] A. Elgart, L. Erdös, B. Schlein, H.-T. Yau, Gross–Pitaevskii equation as the mean field limit of weakly coupled bosons, Arch. Rat. Mech. Anal. 179 no. 2 (2006), 265-283 | MR 2209131 | Zbl 1086.81035

[9] A. Elgart, B. Schlein, Mean field dynamics of boson stars, Comm. Pure Appl. Math. 60 no. 4 (2007), 500-545 | MR 2290709 | Zbl 1113.81032

[10] L. Erdös, B. Schlein, H.-T. Yau, Derivation of the Gross–Pitaevskii hierarchy for the dynamics of Bose–Einstein condensate, Comm. Pure Appl. Math. 59 no. 12 (2006), 1659-1741 | MR 2257859 | Zbl 1122.82018

[11] L. Erdös, B. Schlein, H.-T. Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math. 167 (2007), 515-614 | MR 2276262 | Zbl 1123.35066

[12] L. Erdös, B. Schlein, H.-T. Yau, Derivation of the Gross–Pitaevskii equation for the dynamics of Bose–Einstein condensate, Ann. Math. 172 no. 1 (2010), 291-370 | MR 2680421 | Zbl 1204.82028

[13] L. Erdös, H.-T. Yau, Derivation of the nonlinear Schrödinger equation from a many body Coulomb system, Adv. Theor. Math. Phys. 5 no. 6 (2001), 1169-1205 | MR 1926667 | Zbl 1014.81063

[14] J. Fröhlich, S. Graffi, S. Schwarz, Mean-field- and classical limit of many-body Schrödinger dynamics for bosons, Comm. Math. Phys. 271 no. 3 (2007), 681-697 | MR 2291792 | Zbl 1172.82011

[15] J. Fröhlich, A. Knowles, A. Pizzo, Atomism and quantization, J. Phys. A 40 no. 12 (2007), 3033-3045 | MR 2313859 | Zbl 1115.81071

[16] J. Fröhlich, A. Knowles, S. Schwarz, On the mean-field limit of bosons with Coulomb two-body interaction, Comm. Math. Phys. 288 no. 3 (2009), 1023-1059 | MR 2504864 | Zbl 1177.82016

[17] L. Glangetas, F. Merle, A geometrical approach of existence of blow up solutions in H 1 for nonlinear Schrödinger equation, Rep. No. R95031, Laboratoire d'Analyse Numérique, Univ. Pierre and Marie Curie, 1995.

[18] R. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equation, J. Math. Phys. 18 no. 9 (1977), 1794-1797 | MR 460850 | Zbl 0372.35009

[19] M. Grillakis, M. Machedon, D. Margetis, Second-order corrections to mean field evolution for weakly interacting bosons. I, Comm. Math. Phys. 294 no. 1 (2010), 273-301 | MR 2575484 | Zbl 1208.82030

[20] M. Grillakis, D. Margetis, A priori estimates for many-body Hamiltonian evolution of interacting boson system, J. Hyperbolic Differ. Equ. 5 no. 4 (2008), 857-883 | MR 2475483 | Zbl 1160.35357

[21] K. Hepp, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys. 35 (1974), 265-277 | MR 332046

[22] K. Kirkpatrick, B. Schlein, G. Staffilani, Derivation of the two dimensional nonlinear Schrödinger equation from many body quantum dynamics, arXiv:0808.0505 | MR 2752936 | Zbl 1208.81080

[23] S. Klainerman, M. Machedon, On the uniqueness of solutions to the Gross–Pitaevskii hierarchy, Commun. Math. Phys. 279 no. 1 (2008), 169-185 | MR 2377632 | Zbl 1147.37034

[24] E.H. Lieb, R. Seiringer, Proof of Bose–Einstein condensation for dilute trapped gases, Phys. Rev. Lett. 88 (2002), 170409

[25] E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason, The Mathematics of the Bose Gas and Its Condensation, Birkhäuser (2005) | MR 2143817 | Zbl 1170.82325

[26] E.H. Lieb, R. Seiringer, J. Yngvason, A rigorous derivation of the Gross–Pitaevskii energy functional for a two-dimensional Bose gas, Commun. Math. Phys. 224 (2001) | MR 1868990 | Zbl 0996.82010

[27] H. Nawa, Asymptotic and limiting profiles of blowup solutions of the nonlinear Schrödinger equation with critical power, Comm. Pure Appl. Math. 52 no. 2 (1999), 193-270 | MR 1653454

[28] T. Ogawa, Y. Tsutsumi, Blow-up of H 1 solutions for the one-dimensional nonlinear Schrödinger equations with critical power nonlinearity, Proc. Amer. Math. Soc. 111 no. 2 (1991), 487-496 | MR 1045145 | Zbl 0747.35004

[29] I. Rodnianski, B. Schlein, Quantum fluctuations and rate of convergence towards mean field dynamics, Comm. Math. Phys. 291 no. 1 (2009), 31-61 | MR 2530155 | Zbl 1186.82051

[30] H. Spohn, Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys. 52 no. 3 (1980), 569-615 | MR 578142

[31] M. Tsutsumi, Nonexistence of global solutions to the Cauchy problem for the damped non-linear Schrödinger equations, SIAM J. Math. Anal. 15 (1984), 357-366 | MR 731873 | Zbl 0539.35022

[32] S.N. Vlasov, V.A. Petrishchev, V.I. Talanov, Averaged description of wave beams in linear and nonlinear media (the method of moments), Radiophys. and Quantum Electronics 14 (1971), 1062-1070, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14 (1971), 1353-1363