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.

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 0.

Mots clés : Semiclassical limit, Schrödinger equation, compressible Euler equation.
@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},
     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
DA  - 2004-2005///
PB  - Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://www.numdam.org/item/SEDP_2004-2005____A15_0/
UR  - https://www.ams.org/mathscinet-getitem?mr=2182059
LA  - en
ID  - SEDP_2004-2005____A15_0
ER  - 
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 623940 | Zbl 0477.76059

[Bei92] H. Beirao Da Veiga, Data dependence in the mathematical theory of compressible inviscid fluids, Arch. Rational Mech. Anal., 119 (1992), 109–127. | MR 1176361 | Zbl 0754.76068

[Brenier2000] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Diff. Equations, 25 (2000), 737–754. | MR 1748352 | Zbl 0970.35110

[BG] H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Analysis, TMA, 4 (1980), 677–681. | MR 582536 | Zbl 0451.35023

[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 1134129 | Zbl 0759.35001

[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 1131589 | Zbl 0739.35096

[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 105929

[Grenier98] E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time, Proc. Amer. Math. Soc., 126 (1998), 523–530. | MR 1425123 | Zbl 0910.35115

[JP] C. Josserand and Y. Pomeau, Nonlinear aspects of the theory of Bose-Einstein condensates, Nonlinearity 14 (2000), R25–R62. | MR 1862803 | Zbl 1037.82031

[LL] L. D. Landau and E. M. Lifschits, Fluid Mechanics, Course of Theoretical Physics 6 London-New York, Pregamon Press (1989). | Zbl 0655.76001

[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 1674000 | Zbl 0920.35145

[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 2208290 | Zbl 1079.76016

[LP] P. L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553–618. | MR 1251718 | Zbl 0801.35117

[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 1944031 | Zbl 1040.35076

[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 834481 | Zbl 0612.76082

[Si85] T.  C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys., 101 (1985), 475–485. | MR 815196 | Zbl 0606.76088

[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 1013309 | Zbl 0693.35133

[Wigner] E. Wigner, On the quantum correction for the thermodynamic equivalium, Phys. Rev., 40 (1932), 742-759. | Zbl 0004.38201

[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 1880644 | Zbl 1032.81011

[ZP1] Ping Zhang, Wigner measure and the semiclassical limit of Schrödinger-Poisson Equations, SIAM. J. Math. Anal., 34 (2002), 700–718. | MR 1970889 | Zbl 1032.35132

[ZP2] Ping Zhang, Semiclassical limit of nonlinear Schrödinger equation (II), J. Partial Diff. Eqs., 15 (2002), 83–96. | MR 1909288 | Zbl 1003.35116