# American Institute of Mathematical Sciences

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
 [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] Qiangchang Ju, Hailiang Li, Yong Li, Song Jiang. Quasi-neutral limit of the two-fluid Euler-Poisson system. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1577-1590. doi: 10.3934/cpaa.2010.9.1577 [2] Chunhua Wang, Jing Yang. Positive solutions for a nonlinear Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5461-5504. doi: 10.3934/dcds.2018241 [3] Claude Bardos, François Golse, Peter Markowich, Thierry Paul. On the classical limit of the Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5689-5709. doi: 10.3934/dcds.2015.35.5689 [4] Lihui Chai, Shi Jin, Qin Li. Semi-classical models for the Schrödinger equation with periodic potentials and band crossings. Kinetic & Related Models, 2013, 6 (3) : 505-532. doi: 10.3934/krm.2013.6.505 [5] Zhengping Wang, Huan-Song Zhou. Positive solution for a nonlinear stationary Schrödinger-Poisson system in $R^3$. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 809-816. doi: 10.3934/dcds.2007.18.809 [6] Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025 [7] Zhi Chen, Xianhua Tang, Ning Zhang, Jian Zhang. Standing waves for Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6103-6129. doi: 10.3934/dcds.2019266 [8] Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible Euler-Maxwell equations. Communications on Pure & Applied Analysis, 2013, 12 (1) : 503-518. doi: 10.3934/cpaa.2013.12.503 [9] Xueke Pu. Quasineutral limit of the Euler-Poisson system under strong magnetic fields. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2095-2111. doi: 10.3934/dcdss.2016086 [10] Shu Wang, Chundi Liu. Boundary Layer Problem and Quasineutral Limit of Compressible Euler-Poisson System. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2177-2199. doi: 10.3934/cpaa.2017108 [11] Antonio Azzollini, Pietro d’Avenia, Valeria Luisi. Generalized Schrödinger-Poisson type systems. Communications on Pure & Applied Analysis, 2013, 12 (2) : 867-879. doi: 10.3934/cpaa.2013.12.867 [12] Jiang Xu, Ting Zhang. Zero-electron-mass limit of Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4743-4768. doi: 10.3934/dcds.2013.33.4743 [13] Mingzheng Sun, Jiabao Su, Leiga Zhao. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 427-440. doi: 10.3934/dcds.2015.35.427 [14] Margherita Nolasco. Breathing modes for the Schrödinger-Poisson system with a multiple--well external potential. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1411-1419. doi: 10.3934/cpaa.2010.9.1411 [15] Claudianor O. Alves, Minbo Yang. Existence of positive multi-bump solutions for a Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5881-5910. doi: 10.3934/dcds.2016058 [16] Amna Dabaa, O. Goubet. Long time behavior of solutions to a Schrödinger-Poisson system in $R^3$. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1743-1756. doi: 10.3934/cpaa.2016011 [17] Juntao Sun, Tsung-Fang Wu, Zhaosheng Feng. Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1889-1933. doi: 10.3934/dcds.2018077 [18] Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4685-4702. doi: 10.3934/dcdsb.2018329 [19] Sitong Chen, Junping Shi, Xianhua Tang. Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5867-5889. doi: 10.3934/dcds.2019257 [20] Tadahiro Oh. Global existence for the defocusing nonlinear Schrödinger equations with limit periodic initial data. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1563-1580. doi: 10.3934/cpaa.2015.14.1563

2018 Impact Factor: 1.38