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.  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.  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.  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.   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.  doi: 10.1002/mma.1670140103.  Google Scholar

[6]

F. Castella, "Effects disperifs pour les équations de Vlasov et de Schrödinger,", Ph.D. Thesis, (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.  doi: 10.1142/S0218202597000530.  Google Scholar

[8]

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

[9]

E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time,, Proc. Amer. Math. Soc., 126 (1998), 523.  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.   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.  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.   Google Scholar

[13]

T. Kato, Nonstationary flows of viscous and ideal fluids in $R$$^3$,, J. Funct. Anal., 9 (1972), 296.  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).   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 (1996).   Google Scholar

[17]

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

[18]

A. Majda, "Compressible Fluids Flow and Systems of Conservation Laws in Several Space Variables,", Applied Mathematical Sciences, 53 (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.  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.   Google Scholar

[21]

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

[22]

W. Strauss, "Nonlinear Wave Equations,", CBMS Regional Conference Series in Mathematics, 73 (1989).   Google Scholar

[23]

C. Sulem and P.-L. Sulem, "The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse,", Applied Mathematical Sciences, 139 (1999).   Google Scholar

[24]

P. Zhang, Wigner measure and the semiclassical limit of Schrödinger-Poisson equations,, SIAM J. Math. Anal., 34 (2002), 700.  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.  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.  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.  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.  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.   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.  doi: 10.1002/mma.1670140103.  Google Scholar

[6]

F. Castella, "Effects disperifs pour les équations de Vlasov et de Schrödinger,", Ph.D. Thesis, (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.  doi: 10.1142/S0218202597000530.  Google Scholar

[8]

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

[9]

E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time,, Proc. Amer. Math. Soc., 126 (1998), 523.  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.   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.  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.   Google Scholar

[13]

T. Kato, Nonstationary flows of viscous and ideal fluids in $R$$^3$,, J. Funct. Anal., 9 (1972), 296.  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).   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 (1996).   Google Scholar

[17]

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

[18]

A. Majda, "Compressible Fluids Flow and Systems of Conservation Laws in Several Space Variables,", Applied Mathematical Sciences, 53 (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.  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.   Google Scholar

[21]

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

[22]

W. Strauss, "Nonlinear Wave Equations,", CBMS Regional Conference Series in Mathematics, 73 (1989).   Google Scholar

[23]

C. Sulem and P.-L. Sulem, "The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse,", Applied Mathematical Sciences, 139 (1999).   Google Scholar

[24]

P. Zhang, Wigner measure and the semiclassical limit of Schrödinger-Poisson equations,, SIAM J. Math. Anal., 34 (2002), 700.  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.  doi: 10.1002/cpa.3017.  Google Scholar

[1]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[2]

Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020125

[3]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003

[4]

Jianfeng Huang, Haihua Liang. Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 861-873. doi: 10.3934/dcdsb.2020145

[5]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002

[6]

Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304

[7]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011

[8]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[9]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316

[10]

Masaru Hamano, Satoshi Masaki. A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1415-1447. doi: 10.3934/dcds.2020323

[11]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[12]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[13]

Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328

[14]

Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001

[15]

Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021003

[16]

Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020294

[17]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020298

[18]

Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124

[19]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405

[20]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

2019 Impact Factor: 1.311

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]