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 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.

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.

[1]

Qiangchang Ju, Hailiang Li, Yong Li, Song Jiang. Quasi-neutral limit of the two-fluid Euler-Poisson system. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1577-1590. doi: 10.3934/cpaa.2010.9.1577

[2]

Xiaoming An, Xian Yang. Semi-classical states for fractional Schrödinger equations with magnetic fields and fast decaying potentials. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1649-1672. doi: 10.3934/cpaa.2022038

[3]

Claude Bardos, François Golse, Peter Markowich, Thierry Paul. On the classical limit of the Schrödinger equation. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5689-5709. doi: 10.3934/dcds.2015.35.5689

[4]

Chunhua Wang, Jing Yang. Positive solutions for a nonlinear Schrödinger-Poisson system. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5461-5504. doi: 10.3934/dcds.2018241

[5]

Xueqin Peng, Gao Jia. Existence and asymptotical behavior of positive solutions for the Schrödinger-Poisson system with double quasi-linear terms. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2325-2344. doi: 10.3934/dcdsb.2021134

[6]

Lihui Chai, Shi Jin, Qin Li. Semi-classical models for the Schrödinger equation with periodic potentials and band crossings. Kinetic and Related Models, 2013, 6 (3) : 505-532. doi: 10.3934/krm.2013.6.505

[7]

Jianwei Yang, Dongling Li, Xiao Yang. On the quasineutral limit for the compressible Euler-Poisson equations. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022020

[8]

Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible Euler-Maxwell equations. Communications on Pure and Applied Analysis, 2013, 12 (1) : 503-518. doi: 10.3934/cpaa.2013.12.503

[9]

Zhengping Wang, Huan-Song Zhou. Positive solution for a nonlinear stationary Schrödinger-Poisson system in $R^3$. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 809-816. doi: 10.3934/dcds.2007.18.809

[10]

Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1497-1519. doi: 10.3934/cpaa.2021030

[11]

Yanheng Ding, Xiaojing Dong, Qi Guo. On multiplicity of semi-classical solutions to nonlinear Dirac equations of space-dimension $ n $. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4105-4123. doi: 10.3934/dcds.2021030

[12]

Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025

[13]

Zhi Chen, Xianhua Tang, Ning Zhang, Jian Zhang. Standing waves for Schrödinger-Poisson system with general nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6103-6129. doi: 10.3934/dcds.2019266

[14]

Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1819-1835. doi: 10.3934/dcdss.2021038

[15]

Rong Cheng, Jun Wang. Existence of ground states for Schrödinger-Poisson system with nonperiodic potentials. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2021317

[16]

Xueke Pu. Quasineutral limit of the Euler-Poisson system under strong magnetic fields. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2095-2111. doi: 10.3934/dcdss.2016086

[17]

Shu Wang, Chundi Liu. Boundary Layer Problem and Quasineutral Limit of Compressible Euler-Poisson System. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2177-2199. doi: 10.3934/cpaa.2017108

[18]

Jiang Xu, Ting Zhang. Zero-electron-mass limit of Euler-Poisson equations. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4743-4768. doi: 10.3934/dcds.2013.33.4743

[19]

Antonio Azzollini, Pietro d’Avenia, Valeria Luisi. Generalized Schrödinger-Poisson type systems. Communications on Pure and Applied Analysis, 2013, 12 (2) : 867-879. doi: 10.3934/cpaa.2013.12.867

[20]

Tadahiro Oh. Global existence for the defocusing nonlinear Schrödinger equations with limit periodic initial data. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1563-1580. doi: 10.3934/cpaa.2015.14.1563

2020 Impact Factor: 1.432

Metrics

  • PDF downloads (79)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]