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

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

Received  January 2011 Revised  May 2011 Published  August 2011

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.
Citation: Qiangchang Ju, Fucai Li, Hailiang Li. Asymptotic limit of nonlinear Schrödinger-Poisson system with general initial data. Kinetic & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), 737-754.  Google Scholar

[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.  Google Scholar

[6]

F. Castella, "Effects disperifs pour les équations de Vlasov et de Schrödinger," Ph.D. Thesis, Université Paris 6, 1997. Google Scholar

[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.  Google Scholar

[8]

T. Cazanava, "An Introduction to Nonlinear Schödinger Equations," Testos de Métodos Matemáticos, Rio de Janeiro, 1980. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. De Leo and D. Rial, Well posedness and smoothing effect of Schrödinger-Poisson equation, J. Math. Phys., 48 (2007), 15 pp.  Google Scholar

[15]

H. L. Li and C.-K. Lin, Semiclassical limit and well-poseness of nonlinear Schrödinger-Poisson systems,, Electron. J. Differential Equations, 2003 ().   Google Scholar

[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.  Google Scholar

[17]

P.-L. Lions and T. Paul, Sur les measure de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618.  Google Scholar

[18]

A. Majda, "Compressible Fluids Flow and Systems of Conservation Laws in Several Space Variables," Applied Mathematical Sciences, 53, Springer-Verlag, New York, 1984.  Google Scholar

[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.  Google Scholar

[20]

N. Masmoudi, From Vlasov-Poisson system to the incompressible Euler system, Comm. Partial Differential Equations, 26 (2001), 1913-1928.  Google Scholar

[21]

M. Puel, Convergence of the Schrödinger-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 27 (2002), 2311-2331.  Google Scholar

[22]

W. Strauss, "Nonlinear Wave Equations," CBMS Regional Conference Series in Mathematics, 73, American Mathematical Society, Providence, RI, 1989.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), 737-754.  Google Scholar

[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.  Google Scholar

[6]

F. Castella, "Effects disperifs pour les équations de Vlasov et de Schrödinger," Ph.D. Thesis, Université Paris 6, 1997. Google Scholar

[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.  Google Scholar

[8]

T. Cazanava, "An Introduction to Nonlinear Schödinger Equations," Testos de Métodos Matemáticos, Rio de Janeiro, 1980. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. De Leo and D. Rial, Well posedness and smoothing effect of Schrödinger-Poisson equation, J. Math. Phys., 48 (2007), 15 pp.  Google Scholar

[15]

H. L. Li and C.-K. Lin, Semiclassical limit and well-poseness of nonlinear Schrödinger-Poisson systems,, Electron. J. Differential Equations, 2003 ().   Google Scholar

[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.  Google Scholar

[17]

P.-L. Lions and T. Paul, Sur les measure de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618.  Google Scholar

[18]

A. Majda, "Compressible Fluids Flow and Systems of Conservation Laws in Several Space Variables," Applied Mathematical Sciences, 53, Springer-Verlag, New York, 1984.  Google Scholar

[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.  Google Scholar

[20]

N. Masmoudi, From Vlasov-Poisson system to the incompressible Euler system, Comm. Partial Differential Equations, 26 (2001), 1913-1928.  Google Scholar

[21]

M. Puel, Convergence of the Schrödinger-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 27 (2002), 2311-2331.  Google Scholar

[22]

W. Strauss, "Nonlinear Wave Equations," CBMS Regional Conference Series in Mathematics, 73, American Mathematical Society, Providence, RI, 1989.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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