Wigner measure and the semiclassical limit of Schrödinger-Poisson equations

SIAM Journal on Mathematical Analysis

Volumn 34, Issue 3, 2003, Pages 700-718

  Zhang, Ping ^a  

^a CHINESE ACADEMY OF SCIENCES   (China)

Author keywords

Euler Poisson; Pseudodifferential operator; Schr dinger Poisson; Vlasov Poisson; Wigner measure; Wigner transformation

Indexed keywords

EID: 0038237443     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141001393407     Document Type: Article

References (25)

1. Y. BRENIER, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), pp. 737-754.
2. F. BREZZI AND P. A. MARKOWICH, The three dimensional Wigner-Poisson problem: Existence, uniqueness and approximation, Math. Methods Appl. Sci., 14 (1991), pp. 35-62.
3. B. DESJARDINS, C.-K. LIN, AND T.-C. TSO, Semiclassical limit of the derivative nonlinear Schrödinger equation, Math. Models Methods Appl. Sci., 10 (2000), pp. 261-285.
4. I. GASSER AND P. A. MARKOWICH, Quantum hydrodynamics, Wigner transforms and the classical limit, Asymptot. Anal., 14 (1997), pp. 97-116.
5. I. GASSER, C. K. LIN, AND P. A. MARKOWICH, A review of dispersive limits of (non)linear Schrödinger-type equations, Taiwanese J. Math., 4 (2000), pp. 501-529.
6. P. GÉRARD, Microlocal defect measures, Comm. Partial Differential Equations, 16 (1991), pp. 1761-1794.
7. P. GÉRARD, P. A. MARKOWICH, N. J. MAUSER, AND F. POUPAUD, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), pp. 323-379.
8. J. GINIBRE AND G. VELO, On the global Cauchy problem for some nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), pp. 309-323.
9. E. GRENIER, Semiclassical limit of the nonlinear Schrödinger equation in small time, Proc. Amer. Math. Soc., 126 (1998), pp. 523-530.
10. S. JIN, C. D. LEVERMORE, AND D. W. MCLAUGHLIN, The behavior of solutions of the NLS equation in the semiclassical limit, in Singular Limits of Dispersive Waves, NATO Adv. Sci. Inst. Ser. B Phys. 320, N. Ercolani, I. Gabitov, C. D. Levermore, and D. Serre, eds., Plenum, New York, 1994, pp. 235-256.
11. S. JIN, C. D. LEVERMORE, AND D. W. MCLAUGHLIN, The semiclassical limit of the defocusing NLS hierarchy, Comm. Pure Appl. Math., 52 (1999), pp. 613-654.
12. L. D. LANDAU AND E. M. LIFSCHITZ, Lehrbuch der Theoretischen Physik III. Quantenmechanik, Akademie-Verlag, Berlin, 1985.
13. F. LIN AND P. ZHANG, On the hydrodynamic limit of Ginzburg-Landau wave vortices, Comm. Pure Appl. Math., 55 (2002), pp. 831-856.
14. P. L. LIONS AND T. PAUL, Sur les measures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), pp. 553-618.
15. M. I. LOFFREDO AND L. M. MORATO, Self-consistent hydrodynamical model for Hell near absolute zero in the frame work of stochastic mechanics, Phys. Rev. B, 35 (1987), pp. 1742-1747.
16. A. MAJDA, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Appl. Math. Sci. 53, Springer-Verlag, New York, 1984.
17. P. A. MARKOWICH, C. RINGHOFER, AND C. SCHMEISER, Semiconductor Equations, Springer-Verlag, New York, 1990.
18. M. PUEL, Convergence du système de Schrödinger Poisson vers les ŕquations d'Euler, preprint, 2001.
19. E. M. STEIN AND G. WEISS, Introduction to Fourier Analysis on Euclidean Space, Princeton University Press, Princeton, NJ, 1982.
20. L. TARTAR, H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), pp. 193-230.
21. M. TAYLOR, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser Boston, Boston, MA, 1991.
22. Z. XIN AND P. ZHANG, On the global existence of weak solutions to the shallow water equation, Comm. Pure Appl. Math., 53 (2000), pp. 1411-1433.
23. E. WIGNER, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), pp. 749-759.
24. P. ZHANG AND Y. ZHENG, Existence and uniqueness of solutions of an asymptotic equation arising from a variational wave equation with general data, Arch. Ration. Mech. Anal., 155 (2000), pp. 49-83.
25. P. ZHANG, Y. ZHENG, AND N. J. MAUSER, The limit from Schrödinger-Poisson to Vlasov-Poisson with general data in one dimension, Comm. Pure Appl. Math., 55 (2002), pp. 582-632.