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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 |
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 $\hbar>0$.
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. |
show all references
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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