Asymptotic limit of nonlinear Schrödinger-Poisson system with general initial data

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September 2011, 4(3): 767-783. doi: 10.3934/krm.2011.4.767

Asymptotic limit of nonlinear Schrödinger-Poisson system with general initial data

Qiangchang Ju ^1, , Fucai Li ^2, and Hailiang Li ^3,

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

Received January 2011 Revised May 2011 Published August 2011

Abstract
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The asymptotic limit of the nonlinear Schrödinger-Poisson system with general WKB initial data is studied in this paper. It is proved that the current, defined by the smooth solution of the nonlinear Schrödinger-Poisson system, converges to the strong solution of the incompressible Euler equations plus a term of fast singular oscillating gradient vector fields when both the Planck constant $\hbar$ and the Debye length $\lambda$ tend to zero. The proof involves homogenization techniques, theories of symmetric quasilinear hyperbolic system and elliptic estimates, and the key point is to establish the uniformly bounded estimates with respect to both the Planck constant and the Debye length.

Keywords: semi-classical limit, incompressible Euler equations., quasi-neutral limit, Nonlinear Schrödinger-Poisson System.

Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B4.

Citation: Qiangchang Ju, Fucai Li, Hailiang Li. Asymptotic limit of nonlinear Schrödinger-Poisson system with general initial data. Kinetic and Related Models, 2011, 4 (3) : 767-783. doi: 10.3934/krm.2011.4.767

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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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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