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Asymptotic limit of nonlinear Schrödinger-Poisson system with general initial data
1. | Institute of Applied Physics and Computational Mathematics, Box 8009-28, Beijing 100088, China |
2. | Department of Mathematics, Nanjing University, Nanjing 210093 |
3. | Department of Mathematics and Institute of Mathematics and Interdisciplinary Science, Capital Normal University, Beijing 100037, China |
References:
[1] |
T. Alazard and R. Carles, Semi-classical limit of Schrödinger-Poisson equations in space dimension $n\geq 3$, J. Differential Equations, 233 (2007), 241-275.
doi: 10.1016/j.jde.2006.10.003. |
[2] |
A. Arnold and F. Nier, The two-dimensional Wigner-Poisson problem for an electron gas in the charge neutral case, Math. Methods Appl. Sci., 14 (1991), 595-613.
doi: 10.1002/mma.1670140902. |
[3] |
P. Bechouche, N. J. Mauser and F. Poupaud, Semiclassical limit for the Schrödinger-Poisson equation in a crystal, Comm. Pure Appl. Math., 54 (2001), 852-890.
doi: 10.1002/cpa.3004. |
[4] |
Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), 737-754. |
[5] |
F. Brezzi and P. A. Markowich, The three-dimensional Wigner-Poisson problem: Existence, Uniqueness and Approximation, Math. Methods Appl. Sci., 14 (1991), 35-61.
doi: 10.1002/mma.1670140103. |
[6] |
F. Castella, "Effects disperifs pour les équations de Vlasov et de Schrödinger," Ph.D. Thesis, Université Paris 6, 1997. |
[7] |
F. Castella, $L^2$ solutions to the Schrödinger-Poisson system: existence, uniqueness, time behavior and smoothing effects, Math. Models Methods Appl. Sci., 7 (1997), 1051-1083.
doi: 10.1142/S0218202597000530. |
[8] |
T. Cazanava, "An Introduction to Nonlinear Schödinger Equations," Testos de Métodos Matemáticos, Rio de Janeiro, 1980. |
[9] |
E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time, Proc. Amer. Math. Soc., 126 (1998), 523-530.
doi: 10.1090/S0002-9939-98-04164-1. |
[10] |
C. C. Hao and H. L. Li, On the initial value problem for the bipolar Schrödinger-Poisson systems, J. Partial Differential Equations, 17 (2004), 283-288. |
[11] |
C. C. Hao, L. Hsiao and H. L. Li, Modified scattering for bipolar nonlinear Schrödiner-Poisson equations, Math. Models Methods Appl. Sci., 14 (2004), 1481-1494.
doi: 10.1142/S0218202504003684. |
[12] |
A. Jüngel and S. Wang, Convergence of nonlinear Schrödinger-Poisson system to the compressible Euler equations, Comm. Partial Differential Equations, 28 (2003), 1005-1022. |
[13] |
T. Kato, Nonstationary flows of viscous and ideal fluids in $R$$^3$, J. Funct. Anal., 9 (1972), 296-305.
doi: 10.1016/0022-1236(72)90003-1. |
[14] |
M. De Leo and D. Rial, Well posedness and smoothing effect of Schrödinger-Poisson equation, J. Math. Phys., 48 (2007), 15 pp. |
[15] |
H. L. Li and C.-K. Lin, Semiclassical limit and well-poseness of nonlinear Schrödinger-Poisson systems,, Electron. J. Differential Equations, 2003 ().
|
[16] |
P.-L. Lions, "Mathematical Topics in Fluid Mechanics, Vol. 1: Incompressible Models," Oxford Lecture Series in Mathematics and its Applications, 3, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996. |
[17] |
P.-L. Lions and T. Paul, Sur les measure de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618. |
[18] |
A. Majda, "Compressible Fluids Flow and Systems of Conservation Laws in Several Space Variables," Applied Mathematical Sciences, 53, Springer-Verlag, New York, 1984. |
[19] |
P. A. Markowich and N. J. Mauser, The classical limit of the self-consistent quantum-Vlasov equations in 3D, Math. Models Methods Appl. Sci., 3 (1993), 109-124.
doi: 10.1142/S0218202593000072. |
[20] |
N. Masmoudi, From Vlasov-Poisson system to the incompressible Euler system, Comm. Partial Differential Equations, 26 (2001), 1913-1928. |
[21] |
M. Puel, Convergence of the Schrödinger-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 27 (2002), 2311-2331. |
[22] |
W. Strauss, "Nonlinear Wave Equations," CBMS Regional Conference Series in Mathematics, 73, American Mathematical Society, Providence, RI, 1989. |
[23] |
C. Sulem and P.-L. Sulem, "The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse," Applied Mathematical Sciences, 139, Springer-Verlag, New York, 1999. |
[24] |
P. Zhang, Wigner measure and the semiclassical limit of Schrödinger-Poisson equations, SIAM J. Math. Anal., 34 (2002), 700-718.
doi: 10.1137/S0036141001393407. |
[25] |
P. Zhang, Y.-X Zheng and N. Mauser, The limit from the Schrödinger-Poisson to Vlasov-Poisson equations with general data in one dimension, Comm. Pure Appl. Math., 55 (2002), 582-632.
doi: 10.1002/cpa.3017. |
show all references
References:
[1] |
T. Alazard and R. Carles, Semi-classical limit of Schrödinger-Poisson equations in space dimension $n\geq 3$, J. Differential Equations, 233 (2007), 241-275.
doi: 10.1016/j.jde.2006.10.003. |
[2] |
A. Arnold and F. Nier, The two-dimensional Wigner-Poisson problem for an electron gas in the charge neutral case, Math. Methods Appl. Sci., 14 (1991), 595-613.
doi: 10.1002/mma.1670140902. |
[3] |
P. Bechouche, N. J. Mauser and F. Poupaud, Semiclassical limit for the Schrödinger-Poisson equation in a crystal, Comm. Pure Appl. Math., 54 (2001), 852-890.
doi: 10.1002/cpa.3004. |
[4] |
Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), 737-754. |
[5] |
F. Brezzi and P. A. Markowich, The three-dimensional Wigner-Poisson problem: Existence, Uniqueness and Approximation, Math. Methods Appl. Sci., 14 (1991), 35-61.
doi: 10.1002/mma.1670140103. |
[6] |
F. Castella, "Effects disperifs pour les équations de Vlasov et de Schrödinger," Ph.D. Thesis, Université Paris 6, 1997. |
[7] |
F. Castella, $L^2$ solutions to the Schrödinger-Poisson system: existence, uniqueness, time behavior and smoothing effects, Math. Models Methods Appl. Sci., 7 (1997), 1051-1083.
doi: 10.1142/S0218202597000530. |
[8] |
T. Cazanava, "An Introduction to Nonlinear Schödinger Equations," Testos de Métodos Matemáticos, Rio de Janeiro, 1980. |
[9] |
E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time, Proc. Amer. Math. Soc., 126 (1998), 523-530.
doi: 10.1090/S0002-9939-98-04164-1. |
[10] |
C. C. Hao and H. L. Li, On the initial value problem for the bipolar Schrödinger-Poisson systems, J. Partial Differential Equations, 17 (2004), 283-288. |
[11] |
C. C. Hao, L. Hsiao and H. L. Li, Modified scattering for bipolar nonlinear Schrödiner-Poisson equations, Math. Models Methods Appl. Sci., 14 (2004), 1481-1494.
doi: 10.1142/S0218202504003684. |
[12] |
A. Jüngel and S. Wang, Convergence of nonlinear Schrödinger-Poisson system to the compressible Euler equations, Comm. Partial Differential Equations, 28 (2003), 1005-1022. |
[13] |
T. Kato, Nonstationary flows of viscous and ideal fluids in $R$$^3$, J. Funct. Anal., 9 (1972), 296-305.
doi: 10.1016/0022-1236(72)90003-1. |
[14] |
M. De Leo and D. Rial, Well posedness and smoothing effect of Schrödinger-Poisson equation, J. Math. Phys., 48 (2007), 15 pp. |
[15] |
H. L. Li and C.-K. Lin, Semiclassical limit and well-poseness of nonlinear Schrödinger-Poisson systems,, Electron. J. Differential Equations, 2003 ().
|
[16] |
P.-L. Lions, "Mathematical Topics in Fluid Mechanics, Vol. 1: Incompressible Models," Oxford Lecture Series in Mathematics and its Applications, 3, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996. |
[17] |
P.-L. Lions and T. Paul, Sur les measure de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618. |
[18] |
A. Majda, "Compressible Fluids Flow and Systems of Conservation Laws in Several Space Variables," Applied Mathematical Sciences, 53, Springer-Verlag, New York, 1984. |
[19] |
P. A. Markowich and N. J. Mauser, The classical limit of the self-consistent quantum-Vlasov equations in 3D, Math. Models Methods Appl. Sci., 3 (1993), 109-124.
doi: 10.1142/S0218202593000072. |
[20] |
N. Masmoudi, From Vlasov-Poisson system to the incompressible Euler system, Comm. Partial Differential Equations, 26 (2001), 1913-1928. |
[21] |
M. Puel, Convergence of the Schrödinger-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 27 (2002), 2311-2331. |
[22] |
W. Strauss, "Nonlinear Wave Equations," CBMS Regional Conference Series in Mathematics, 73, American Mathematical Society, Providence, RI, 1989. |
[23] |
C. Sulem and P.-L. Sulem, "The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse," Applied Mathematical Sciences, 139, Springer-Verlag, New York, 1999. |
[24] |
P. Zhang, Wigner measure and the semiclassical limit of Schrödinger-Poisson equations, SIAM J. Math. Anal., 34 (2002), 700-718.
doi: 10.1137/S0036141001393407. |
[25] |
P. Zhang, Y.-X Zheng and N. Mauser, The limit from the Schrödinger-Poisson to Vlasov-Poisson equations with general data in one dimension, Comm. Pure Appl. Math., 55 (2002), 582-632.
doi: 10.1002/cpa.3017. |
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