In this paper, we study the semiclassical limit of the cubic nonlinear Schrödinger equation with the Neumann boundary condition in an exterior domain. We prove that before the formation of singularities in the limit system, the quantum density and the quantum momentum converge to the unique solution of the compressible Euler equation with the slip boundary condition as the scaling parameter approaches
@article{SEDP_2004-2005____A15_0,
author = {Lin, Fanghua and Zhang, Ping},
title = {Semiclassical {Limit} of the cubic nonlinear {Schr\"odinger} {Equation} concerning a superfluid passing an obstacle},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
note = {talk:15},
pages = {1--13},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2004-2005},
mrnumber = {2182059},
language = {en},
url = {http://www.numdam.org/item/SEDP_2004-2005____A15_0/}
}
TY - JOUR AU - Lin, Fanghua AU - Zhang, Ping TI - Semiclassical Limit of the cubic nonlinear Schrödinger Equation concerning a superfluid passing an obstacle JO - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:15 PY - 2004-2005 SP - 1 EP - 13 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://www.numdam.org/item/SEDP_2004-2005____A15_0/ LA - en ID - SEDP_2004-2005____A15_0 ER -
%0 Journal Article %A Lin, Fanghua %A Zhang, Ping %T Semiclassical Limit of the cubic nonlinear Schrödinger Equation concerning a superfluid passing an obstacle %J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" %Z talk:15 %D 2004-2005 %P 1-13 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U http://www.numdam.org/item/SEDP_2004-2005____A15_0/ %G en %F SEDP_2004-2005____A15_0
Lin, Fanghua; Zhang, Ping. Semiclassical Limit of the cubic nonlinear Schrödinger Equation concerning a superfluid passing an obstacle. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2004-2005), Exposé no. 15, 13 p. http://www.numdam.org/item/SEDP_2004-2005____A15_0/
[Bei81] H. Beirao Da Veiga, On the barotropic motion of compressible perfect fluids, Ann. Sc. Norm. Sup. Pisa, 8 (1981), 417–351. | Numdam | MR | Zbl
[Bei92] H. Beirao Da Veiga, Data dependence in the mathematical theory of compressible inviscid fluids, Arch. Rational Mech. Anal., 119 (1992), 109–127. | MR | Zbl
[Brenier2000] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Diff. Equations, 25 (2000), 737–754. | MR | Zbl
[BG] H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Analysis, TMA, 4 (1980), 677–681. | MR | Zbl
[DD] G. C. Dong, Nonlinear second order parabolic equations. Translated from the Chinese by Kai Seng Chou [Kaising Tso]. Translations of Mathematical Monographs, 95. American Mathematical Society, Providence, RI, 1991. viii+251 pp . | MR | Zbl
[FPR] T. Frisch, Y. Pomeau and S. Rica, Transition to dissipation in a model of superflow, Phys. Rev. Lett. 69, No. 11, (1992), 1644–1647.
[Gerard] P. Gérard, Mesures semi-classiques et ondes de Bloch, Séminaire sur les équations aux Dérivées Partielles, 1990–1991, Exp. No. XVI, 19 École Polytech., Palaiseau, 1991. | Numdam | MR | Zbl
[G] E. P. Gross, J. Math. Phys. 4 (1963), 195–
[GP] V. L. Ginzburg and L. P. Pitaevskii, On the theory of superfluidity, Sov. Phys. JETP 7 (1958), 585. | MR
[Grenier98] E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time, Proc. Amer. Math. Soc., 126 (1998), 523–530. | MR | Zbl
[JP] C. Josserand and Y. Pomeau, Nonlinear aspects of the theory of Bose-Einstein condensates, Nonlinearity 14 (2000), R25–R62. | MR | Zbl
[LL] L. D. Landau and E. M. Lifschits, Fluid Mechanics, Course of Theoretical Physics 6 London-New York, Pregamon Press (1989). | Zbl
[LX] F. H. Lin and J. Xin, On the Incompressible Fluid Limit and the Vortex Motion Law of the Nonlinear Schrödinger Equation, Comm. Math. Phys. 200 (1999), 249–274. | MR | Zbl
[LZ] F. H. Lin and Ping Zhang, Semiclassical limit of the Gross-Pitaevskii equation in an exterior domain, Arch. Rational Mech. Anal., (to appear). | MR | Zbl
[LP] P. L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553–618. | MR | Zbl
[M] E. Madelung, Quanten theorie in Hydrodynamic Form, Z. Physik 40 (1927), 322.
[MP1] M. Puel, Convergence of the Schrödinger-Poisson system to the incompressible Euler equations, Comm. Partial Diff. Equations, 27 (2002), 2311–2331. | MR | Zbl
[Sch86] S. Schochet, The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit, Comm. Math. Phys., 104 (1986), 49–75. | MR | Zbl
[Si85] T. C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys., 101 (1985), 475–485. | MR | Zbl
[TS] M. Tsutsumi, On smooth solutions to the initial-boundary value problem for the nonlinear Schrödinger equation in two space diemnsion, Nonlinear Analysis, TMA, 13 (1989), 1051–1056. | MR | Zbl
[Wigner] E. Wigner, On the quantum correction for the thermodynamic equivalium, Phys. Rev., 40 (1932), 742-759. | Zbl
[ZZM] Ping Zhang, Yuxi Zheng, and N. J. Mauser, The limit from the Schrödinger-Poisson to the Vlasov-Poisson equations with general data in one dimension, Comm. Pure Appl. Math., 55 (2002), 582–632. | MR | Zbl
[ZP1] Ping Zhang, Wigner measure and the semiclassical limit of Schrödinger-Poisson Equations, SIAM. J. Math. Anal., 34 (2002), 700–718. | MR | Zbl
[ZP2] Ping Zhang, Semiclassical limit of nonlinear Schrödinger equation (II), J. Partial Diff. Eqs., 15 (2002), 83–96. | MR | Zbl
