# American Institute of Mathematical Sciences

January  2017, 16(1): 273-294. doi: 10.3934/cpaa.2017013

## Quasineutral limit for the quantum Navier-Stokes-Poisson equations

 1 School of Mathematics and Statistics, Chongqing University, Chongqing 401331, China 2 Department of Mathematics, Chongqing University, Chongqing 401331, China 3 College of Applied Sciences, Beijing University of Technology, Beijing 100124, China

Received  May 2016 Revised  September 2016 Published  September 2016

Fund Project: The second author is supported by NSFC (grant 11471057) and the Fundamental Research Funds for the Central Universities (grant Project No. 106112016CDJZR105501). The third author is supported by NSFC (grant 11371042) and the key fundation of Beijing Municipal Education Commission.

In this paper, we study the quasineutral limit and asymptotic behaviors for the quantum Navier-Stokes-Possion equation. We apply a formal expansion according to Debye length and derive the neutral incompressible Navier-Stokes equation. To establish this limit mathematically rigorously, we derive uniform (in Debye length) estimates for the remainders, for well-prepared initial data. It is demonstrated that the quantum effect do play important roles in the estimates and the norm introduced depends on the Planck constant $\hbar>0$.

Citation: Min Li, Xueke Pu, Shu Wang. Quasineutral limit for the quantum Navier-Stokes-Poisson equations. Communications on Pure and Applied Analysis, 2017, 16 (1) : 273-294. doi: 10.3934/cpaa.2017013
##### References:
 [1] D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden" valuables, Ⅰ, Physical Review, 85 (1952), 166-179. [2] D. Bresch, B. Desjardins and B. Ducomet, Quasi-neutral limit for a viscous capillary model of plasma, Ann. Inst. H. Poincaré Anal. Nonlinear, 22 (2005), 1-9. doi: 10.1016/j.anihpc.2004.02.001. [3] S. Brull and F. Méhats, Derivation of viscous correction terms for the isothermal quantum Euler model, Z. Angew. Math. Mech., 90 (2010), 219-230. doi: 10.1002/zamm.200900297. [4] L. Chen, D. Donatelli and P. Marcati, Incompressible type limit analysis of a hydrodynamic model for charge-carrier transport, SIAM Journal on Mathematical Analysis, 45 (2013), 915-933. doi: 10.1137/120876630. [5] P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press, 1988. [6] S. Cordier and E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics, Commun. Partial Differential Equations, 23 (2000), 1099-1113. doi: 10.1080/03605300008821542. [7] P. Degond, S. Gallego and F. Méhats, Isothermal quantum hydrodynamics: derivation, asymptotic analysis, and simulation, Multiscale Model. Simul., 6 (2007), 246-272. doi: 10.1137/06067153X. [8] P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum hydrodynamic and diffusion models derived from the entropy principle, Quantum Transport, Springer Berlin Heidelberg, (2008), 111-168. doi: 10.1007/978-3-540-79574-2_3. [9] P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum diffusion models derived from the entropy principle, Progress in Industrial Mathematics at ECMI 2006, Springer Berlin Heidelberg, (2008), 106-122. doi: 10.1007/978-3-540-71992-2_6. [10] P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118 (2005), 625-665. doi: 10.1007/s10955-004-8823-3. [11] P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (2003), 587-628. doi: 10.1023/A:1023824008525. [12] D. Donatelli and P. Marcati, A quasineutral type limit for the Navier-Stokes-Poisson system with large data, Nonlinearity, 21 (2008), 135-148. doi: 10.1088/0951-7715/21/1/008. [13] D. Donatelli and P. Marcati, Analysis of oscillations and defect measures for the quasineutral limit in plasma physics, Arch. Ration. Mech. Anal., 206 (2012), 159-188. doi: 10.1007/s00205-012-0531-6. [14] D. Donatelli and P. Marcati, Quasineutral limit, dispersion and oscillations for Korteweg type fluids, SIAM J. Math. Anal. , 47, (2015), 2265-2282. doi: 10.1137/140987651. [15] J. E. Dunn and J. Serrin, On the thermodynamics of interstitial working, Arch. Ration. Mech. Anal., 88 (1985), 95-133. doi: 10.1007/BF00250907. [16] C. L. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54 (1994), 409-427. doi: 10.1137/S0036139992240425. [17] I. Gasser and P. Marcati, A vanishing Debye length limit in a hydrodynamic model for semiconductors, Hyperbolic Problems: Theory, Numerics, Applications, (2001), 409-414. [18] I. Gasser and P. Marcati, The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors, Math. Methods Appl. Sci., 24 (2001), 81-92. doi: 10.1002/1099-1476(20010125)24:2<81::AID-MMA198>3.3.CO;2-O. [19] F. Haas, Quantum Plasmas: An Hydrodynamic Approach, Springer, New York, 2011. doi: 10.1007/978-1-4419-8201-8. [20] H. Hattori and D. Li, Solutions for two dimensional system for materials of Korteweg type, SIAM J. Math. Anal., 25 (1994), 85-98. doi: 10.1137/S003614109223413X. [21] H. Hattori and D. Li, Global solutions of a high dimensional system for Korteweg materials, J. Math. Anal. Appl., 198 (1996), 84-97. doi: 10.1006/jmaa.1996.0069. [22] Q. Ju, F. Li and H. Li, The quasineutral limit of compressible Navier-Stokes-Poisson system with heat conductivity and general initial data, J. Differential Equations, 247 (2009), 203-224. doi: 10.1016/j.jde.2009.02.019. [23] A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045. doi: 10.1137/090776068. [24] A. Jüngel, C.-K. Lin and K.-C. Wu, An asymptotic limit of a Navier-Stokes system with capillary effects, Comm. Math. Phys., 329 (2014), 725-744. doi: 10.1007/s00220-014-1961-9. [25] A. Jüngel and J.-P. Miliŝić, Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solutions, Kinetic and Related Models, 4 (2011), 785-807. doi: 10.3934/krm.2011.4.785. [26] H. Li and C. Lin, Zero Debye length asymptotic of the quantum hydrodynamic model for semiconductors, Comm. Math. Phys., 256 (2005), 195-212. doi: 10.1007/s00220-005-1316-7. [27] Y. Li and W. Yong, Quasi-neutral limit in a 3D compressible Navier-Stokes-Poisson-Korteweg model, IMA J. Appl. Math., 80 (2015), 712-727. doi: 10.1093/imamat/hxu008. [28] P. -L. Lions, Mathematical Topics in Fluid Mechanics, vol. 1: Incompressible Models, Oxford Lecture Ser. Math. Appl. , vol. 3, The Clarendon Press/Oxford University Press, New York, 1996. [29] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104. [30] X. Pu and B. Guo, Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction, Kinetic & Related Models, 9 (2016), 165-191. doi: 10.3934/krm.2016.9.165. [31] X. Pu and B. Guo, Quasineutral limit of the pressureless Euler-Poisson equation for ions, Quart. Appl. Math., 74 (2016), 245-273. doi: 10.1090/qam/1424. [32] Y. Peng, Y. Wang and W. Yong, Quasi-neutral limit of the non-isentropic Euler-Poisson system, Proceedings of the Royal Society of Edinburgh, 136A (2006), 1013-1026. doi: 10.1017/S0308210500004856. [33] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton university press, 1970. [34] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, American Mathematical Society, 2001. doi: 10.1090/chel/343. [35] S. Wang, Quasineutral limit of Euler-Poisson system with and without viscosity, Comm. Partial Differential Equations, 29 (2004), 419-456. doi: 10.1081/PDE-120030403. [36] S. Wang and S. Jiang, The convergence of Navier-Stokes-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 31 (2006), 571-591. doi: 10.1080/03605300500361487. [37] E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759.

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##### References:
 [1] D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden" valuables, Ⅰ, Physical Review, 85 (1952), 166-179. [2] D. Bresch, B. Desjardins and B. Ducomet, Quasi-neutral limit for a viscous capillary model of plasma, Ann. Inst. H. Poincaré Anal. Nonlinear, 22 (2005), 1-9. doi: 10.1016/j.anihpc.2004.02.001. [3] S. Brull and F. Méhats, Derivation of viscous correction terms for the isothermal quantum Euler model, Z. Angew. Math. Mech., 90 (2010), 219-230. doi: 10.1002/zamm.200900297. [4] L. Chen, D. Donatelli and P. Marcati, Incompressible type limit analysis of a hydrodynamic model for charge-carrier transport, SIAM Journal on Mathematical Analysis, 45 (2013), 915-933. doi: 10.1137/120876630. [5] P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press, 1988. [6] S. Cordier and E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics, Commun. Partial Differential Equations, 23 (2000), 1099-1113. doi: 10.1080/03605300008821542. [7] P. Degond, S. Gallego and F. Méhats, Isothermal quantum hydrodynamics: derivation, asymptotic analysis, and simulation, Multiscale Model. Simul., 6 (2007), 246-272. doi: 10.1137/06067153X. [8] P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum hydrodynamic and diffusion models derived from the entropy principle, Quantum Transport, Springer Berlin Heidelberg, (2008), 111-168. doi: 10.1007/978-3-540-79574-2_3. [9] P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum diffusion models derived from the entropy principle, Progress in Industrial Mathematics at ECMI 2006, Springer Berlin Heidelberg, (2008), 106-122. doi: 10.1007/978-3-540-71992-2_6. [10] P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118 (2005), 625-665. doi: 10.1007/s10955-004-8823-3. [11] P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (2003), 587-628. doi: 10.1023/A:1023824008525. [12] D. Donatelli and P. Marcati, A quasineutral type limit for the Navier-Stokes-Poisson system with large data, Nonlinearity, 21 (2008), 135-148. doi: 10.1088/0951-7715/21/1/008. [13] D. Donatelli and P. Marcati, Analysis of oscillations and defect measures for the quasineutral limit in plasma physics, Arch. Ration. Mech. Anal., 206 (2012), 159-188. doi: 10.1007/s00205-012-0531-6. [14] D. Donatelli and P. Marcati, Quasineutral limit, dispersion and oscillations for Korteweg type fluids, SIAM J. Math. Anal. , 47, (2015), 2265-2282. doi: 10.1137/140987651. [15] J. E. Dunn and J. Serrin, On the thermodynamics of interstitial working, Arch. Ration. Mech. Anal., 88 (1985), 95-133. doi: 10.1007/BF00250907. [16] C. L. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54 (1994), 409-427. doi: 10.1137/S0036139992240425. [17] I. Gasser and P. Marcati, A vanishing Debye length limit in a hydrodynamic model for semiconductors, Hyperbolic Problems: Theory, Numerics, Applications, (2001), 409-414. [18] I. Gasser and P. Marcati, The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors, Math. Methods Appl. Sci., 24 (2001), 81-92. doi: 10.1002/1099-1476(20010125)24:2<81::AID-MMA198>3.3.CO;2-O. [19] F. Haas, Quantum Plasmas: An Hydrodynamic Approach, Springer, New York, 2011. doi: 10.1007/978-1-4419-8201-8. [20] H. Hattori and D. Li, Solutions for two dimensional system for materials of Korteweg type, SIAM J. Math. Anal., 25 (1994), 85-98. doi: 10.1137/S003614109223413X. [21] H. Hattori and D. Li, Global solutions of a high dimensional system for Korteweg materials, J. Math. Anal. Appl., 198 (1996), 84-97. doi: 10.1006/jmaa.1996.0069. [22] Q. Ju, F. Li and H. Li, The quasineutral limit of compressible Navier-Stokes-Poisson system with heat conductivity and general initial data, J. Differential Equations, 247 (2009), 203-224. doi: 10.1016/j.jde.2009.02.019. [23] A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045. doi: 10.1137/090776068. [24] A. Jüngel, C.-K. Lin and K.-C. Wu, An asymptotic limit of a Navier-Stokes system with capillary effects, Comm. Math. Phys., 329 (2014), 725-744. doi: 10.1007/s00220-014-1961-9. [25] A. Jüngel and J.-P. Miliŝić, Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solutions, Kinetic and Related Models, 4 (2011), 785-807. doi: 10.3934/krm.2011.4.785. [26] H. Li and C. Lin, Zero Debye length asymptotic of the quantum hydrodynamic model for semiconductors, Comm. Math. Phys., 256 (2005), 195-212. doi: 10.1007/s00220-005-1316-7. [27] Y. Li and W. Yong, Quasi-neutral limit in a 3D compressible Navier-Stokes-Poisson-Korteweg model, IMA J. Appl. Math., 80 (2015), 712-727. doi: 10.1093/imamat/hxu008. [28] P. -L. Lions, Mathematical Topics in Fluid Mechanics, vol. 1: Incompressible Models, Oxford Lecture Ser. Math. Appl. , vol. 3, The Clarendon Press/Oxford University Press, New York, 1996. [29] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104. [30] X. Pu and B. Guo, Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction, Kinetic & Related Models, 9 (2016), 165-191. doi: 10.3934/krm.2016.9.165. [31] X. Pu and B. Guo, Quasineutral limit of the pressureless Euler-Poisson equation for ions, Quart. Appl. Math., 74 (2016), 245-273. doi: 10.1090/qam/1424. [32] Y. Peng, Y. Wang and W. Yong, Quasi-neutral limit of the non-isentropic Euler-Poisson system, Proceedings of the Royal Society of Edinburgh, 136A (2006), 1013-1026. doi: 10.1017/S0308210500004856. [33] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton university press, 1970. [34] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, American Mathematical Society, 2001. doi: 10.1090/chel/343. [35] S. Wang, Quasineutral limit of Euler-Poisson system with and without viscosity, Comm. Partial Differential Equations, 29 (2004), 419-456. doi: 10.1081/PDE-120030403. [36] S. Wang and S. Jiang, The convergence of Navier-Stokes-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 31 (2006), 571-591. doi: 10.1080/03605300500361487. [37] E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759.
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