-
Previous Article
Properties of the steady state distribution of electrons in semiconductors
- KRM Home
- This Issue
-
Next Article
Asymptotic limit of nonlinear Schrödinger-Poisson system with general initial data
Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution
1. | Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8-10, 1040 Wien, Austria |
2. | Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb Unska 3, 10000 Zagreb, Croatia |
References:
[1] |
M. Anile, V. Romano and G. Russo, Extended hydrodynamical model of carrier transport in semiconductors, SIAM J. Appl. Math., 61 (2000), 74-101.
doi: 10.1137/S003613999833294X. |
[2] |
G. Baccarani and M. Wordeman, An investigation of steady-state velocity overshoot effects in Si and GaAs devices, Solid-State Electronics, 28 (1985), 407-416.
doi: 10.1016/0038-1101(85)90100-5. |
[3] |
P. Bhatnagar, E. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.
doi: 10.1103/PhysRev.94.511. |
[4] |
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. |
[5] |
A. Caldeira and A. Leggett, Path integral approach to quantum Brownian motion, Physica A, 121 (1983), 587-616.
doi: 10.1016/0378-4371(83)90013-4. |
[6] |
P. Degond, S. Gallego and F. Méhats, On quantum hydrodynamic and quantum energy transport models, Commun. Math. Sci., 5 (2007), 887-908. |
[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, in "Quantum Transport" (eds. G. Allaire et al.), 111-168, Lecture Notes Math., 1946, Springer, Berlin, 2008. |
[9] |
P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum diffusion models derived from the entropy principle, in "Progress in Industrial Mathematics at ECMI 2006" (eds. L. Bonilla et al.), 106-122, Mathematics in Industry, 12, Springer, Berlin, 2008. |
[10] |
P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118 (2005), 625-667.
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] |
J. Dong, A note on barotropic compressible quantum Navier-Stokes equations, Nonlin. Anal., 73 (2010), 854-856.
doi: 10.1016/j.na.2010.03.047. |
[13] |
W. Dreyer, Maximisation of the entropy in non-equilibrium, J. Phys. A, 20 (1987), 6505-6517.
doi: 10.1088/0305-4470/20/18/047. |
[14] |
E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004. |
[15] |
C. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54 (1994), 409-427.
doi: 10.1137/S0036139992240425. |
[16] |
F. Jiang, A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations, Nonlin. Anal. Real World Appl., 12 (2011), 1733-1735.
doi: 10.1016/j.nonrwa.2010.11.005. |
[17] |
A. Jüngel, "Transport Equations for Semiconductors," Lecture Notes in Physics, 773, Springer-Verlag, Berlin, 2009. |
[18] |
A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045. |
[19] |
A. Jüngel, Effective velocity in compressible Navier-Stokes equations with third-order derivatives, Nonlin. Anal., 74 (2011), 2813-2818.
doi: 10.1016/j.na.2011.01.002. |
[20] |
A. Jüngel, Dissipative quantum fluid models, to appear in Revista Mat. Univ. Parma, 2011. |
[21] |
A. Jüngel and D. Matthes, Derivation of the isothermal quantum hydrodynamic equations using entropy minimization, Z. Angew. Math. Mech., 85 (2005), 806-814.
doi: 10.1002/zamm.200510232. |
[22] |
A. Jüngel, D. Matthes and J.-P. Milišić, Derivation of new quantum hydrodynamic equations using entropy minimization, SIAM J. Appl. Math., 67 (2006), 46-68.
doi: 10.1137/050644823. |
[23] |
C. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.
doi: 10.1007/BF02179552. |
[24] |
T. Li, Z. Yu, Y. Wang, L. Huang and C. Xiang, "Numerical Simulation of Negative Differential Resistance Characteristics in Si/Si$_1-x$Ge$_x$ RTD at Room Temperature," Proceedings of the 2005 IEEE Conference on Electron Devices and Solid-State Circuits, (2005), 409-412. |
[25] |
F. Méhats and O. Pinaud, An inverse problem in quantum statistical physics, J. Stat. Phys., 140 (2010), 565–-602.
doi: 10.1007/s10955-010-0003-z. |
show all references
References:
[1] |
M. Anile, V. Romano and G. Russo, Extended hydrodynamical model of carrier transport in semiconductors, SIAM J. Appl. Math., 61 (2000), 74-101.
doi: 10.1137/S003613999833294X. |
[2] |
G. Baccarani and M. Wordeman, An investigation of steady-state velocity overshoot effects in Si and GaAs devices, Solid-State Electronics, 28 (1985), 407-416.
doi: 10.1016/0038-1101(85)90100-5. |
[3] |
P. Bhatnagar, E. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.
doi: 10.1103/PhysRev.94.511. |
[4] |
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. |
[5] |
A. Caldeira and A. Leggett, Path integral approach to quantum Brownian motion, Physica A, 121 (1983), 587-616.
doi: 10.1016/0378-4371(83)90013-4. |
[6] |
P. Degond, S. Gallego and F. Méhats, On quantum hydrodynamic and quantum energy transport models, Commun. Math. Sci., 5 (2007), 887-908. |
[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, in "Quantum Transport" (eds. G. Allaire et al.), 111-168, Lecture Notes Math., 1946, Springer, Berlin, 2008. |
[9] |
P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum diffusion models derived from the entropy principle, in "Progress in Industrial Mathematics at ECMI 2006" (eds. L. Bonilla et al.), 106-122, Mathematics in Industry, 12, Springer, Berlin, 2008. |
[10] |
P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118 (2005), 625-667.
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] |
J. Dong, A note on barotropic compressible quantum Navier-Stokes equations, Nonlin. Anal., 73 (2010), 854-856.
doi: 10.1016/j.na.2010.03.047. |
[13] |
W. Dreyer, Maximisation of the entropy in non-equilibrium, J. Phys. A, 20 (1987), 6505-6517.
doi: 10.1088/0305-4470/20/18/047. |
[14] |
E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004. |
[15] |
C. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54 (1994), 409-427.
doi: 10.1137/S0036139992240425. |
[16] |
F. Jiang, A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations, Nonlin. Anal. Real World Appl., 12 (2011), 1733-1735.
doi: 10.1016/j.nonrwa.2010.11.005. |
[17] |
A. Jüngel, "Transport Equations for Semiconductors," Lecture Notes in Physics, 773, Springer-Verlag, Berlin, 2009. |
[18] |
A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045. |
[19] |
A. Jüngel, Effective velocity in compressible Navier-Stokes equations with third-order derivatives, Nonlin. Anal., 74 (2011), 2813-2818.
doi: 10.1016/j.na.2011.01.002. |
[20] |
A. Jüngel, Dissipative quantum fluid models, to appear in Revista Mat. Univ. Parma, 2011. |
[21] |
A. Jüngel and D. Matthes, Derivation of the isothermal quantum hydrodynamic equations using entropy minimization, Z. Angew. Math. Mech., 85 (2005), 806-814.
doi: 10.1002/zamm.200510232. |
[22] |
A. Jüngel, D. Matthes and J.-P. Milišić, Derivation of new quantum hydrodynamic equations using entropy minimization, SIAM J. Appl. Math., 67 (2006), 46-68.
doi: 10.1137/050644823. |
[23] |
C. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.
doi: 10.1007/BF02179552. |
[24] |
T. Li, Z. Yu, Y. Wang, L. Huang and C. Xiang, "Numerical Simulation of Negative Differential Resistance Characteristics in Si/Si$_1-x$Ge$_x$ RTD at Room Temperature," Proceedings of the 2005 IEEE Conference on Electron Devices and Solid-State Circuits, (2005), 409-412. |
[25] |
F. Méhats and O. Pinaud, An inverse problem in quantum statistical physics, J. Stat. Phys., 140 (2010), 565–-602.
doi: 10.1007/s10955-010-0003-z. |
[1] |
Vincent Giovangigli, Wen-An Yong. Volume viscosity and internal energy relaxation: Symmetrization and Chapman-Enskog expansion. Kinetic and Related Models, 2015, 8 (1) : 79-116. doi: 10.3934/krm.2015.8.79 |
[2] |
Guangwu Wang, Boling Guo. Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6141-6166. doi: 10.3934/dcdsb.2019133 |
[3] |
Xulong Qin, Zheng-An Yao, Hongxing Zhao. One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries. Communications on Pure and Applied Analysis, 2008, 7 (2) : 373-381. doi: 10.3934/cpaa.2008.7.373 |
[4] |
Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041 |
[5] |
Vincent Giovangigli, Wen-An Yong. Erratum: ``Volume viscosity and internal energy relaxation: Symmetrization and Chapman-Enskog expansion''. Kinetic and Related Models, 2016, 9 (4) : 813-813. doi: 10.3934/krm.2016018 |
[6] |
Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163 |
[7] |
Jianwei Yang, Peng Cheng, Yudong Wang. Asymptotic limit of a Navier-Stokes-Korteweg system with density-dependent viscosity. Electronic Research Announcements, 2015, 22: 20-31. doi: 10.3934/era.2015.22.20 |
[8] |
Céline Baranger, Marzia Bisi, Stéphane Brull, Laurent Desvillettes. On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases. Kinetic and Related Models, 2018, 11 (4) : 821-858. doi: 10.3934/krm.2018033 |
[9] |
Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic and Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004 |
[10] |
Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222 |
[11] |
Wenjun Wang, Lei Yao. Spherically symmetric Navier-Stokes equations with degenerate viscosity coefficients and vacuum. Communications on Pure and Applied Analysis, 2010, 9 (2) : 459-481. doi: 10.3934/cpaa.2010.9.459 |
[12] |
Jishan Fan, Tohru Ozawa. An approximation model for the density-dependent magnetohydrodynamic equations. Conference Publications, 2013, 2013 (special) : 207-216. doi: 10.3934/proc.2013.2013.207 |
[13] |
Enrique Fernández-Cara. Motivation, analysis and control of the variable density Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2012, 5 (6) : 1021-1090. doi: 10.3934/dcdss.2012.5.1021 |
[14] |
Quansen Jiu, Zhouping Xin. The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients. Kinetic and Related Models, 2008, 1 (2) : 313-330. doi: 10.3934/krm.2008.1.313 |
[15] |
Jishan Fan, Tohru Ozawa. A regularity criterion for 3D density-dependent MHD system with zero viscosity. Conference Publications, 2015, 2015 (special) : 395-399. doi: 10.3934/proc.2015.0395 |
[16] |
Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure and Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001 |
[17] |
Wuming Li, Xiaojun Liu, Quansen Jiu. The decay estimates of solutions for 1D compressible flows with density-dependent viscosity coefficients. Communications on Pure and Applied Analysis, 2013, 12 (2) : 647-661. doi: 10.3934/cpaa.2013.12.647 |
[18] |
J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647 |
[19] |
Ping Chen, Ting Zhang. A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients. Communications on Pure and Applied Analysis, 2008, 7 (4) : 987-1016. doi: 10.3934/cpaa.2008.7.987 |
[20] |
Yuming Qin, Lan Huang, Shuxian Deng, Zhiyong Ma, Xiaoke Su, Xinguang Yang. Interior regularity of the compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 163-192. doi: 10.3934/dcdss.2009.2.163 |
2021 Impact Factor: 1.398
Tools
Metrics
Other articles
by authors
[Back to Top]