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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.
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.
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.
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.
doi: 10.1002/zamm.200900297. |
[5] |
A. Caldeira and A. Leggett, Path integral approach to quantum Brownian motion,, Physica A, 121 (1983), 587.
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.
|
[7] |
P. Degond, S. Gallego and F. Méhats, Isothermal quantum hydrodynamics: derivation, asymptotic analysis, and simulation,, Multiscale Model. Simul., 6 (2007), 246.
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, 1946 (2008), 111.
|
[9] |
P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum diffusion models derived from the entropy principle,, in, 12 (2006), 106.
|
[10] |
P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models,, J. Stat. Phys., 118 (2005), 625.
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.
doi: 10.1023/A:1023824008525. |
[12] |
J. Dong, A note on barotropic compressible quantum Navier-Stokes equations,, Nonlin. Anal., 73 (2010), 854.
doi: 10.1016/j.na.2010.03.047. |
[13] |
W. Dreyer, Maximisation of the entropy in non-equilibrium,, J. Phys. A, 20 (1987), 6505.
doi: 10.1088/0305-4470/20/18/047. |
[14] |
E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford Series in Mathematics and its Applications, 26 (2004).
|
[15] |
C. Gardner, The quantum hydrodynamic model for semiconductor devices,, SIAM J. Appl. Math., (1994), 409.
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.
doi: 10.1016/j.nonrwa.2010.11.005. |
[17] |
A. Jüngel, "Transport Equations for Semiconductors,", Lecture Notes in Physics, 773 (2009).
|
[18] |
A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids,, SIAM J. Math. Anal., 42 (2010), 1025.
|
[19] |
A. Jüngel, Effective velocity in compressible Navier-Stokes equations with third-order derivatives,, Nonlin. Anal., 74 (2011), 2813.
doi: 10.1016/j.na.2011.01.002. |
[20] |
A. Jüngel, Dissipative quantum fluid models,, to appear in Revista Mat. Univ. Parma, (2011). Google Scholar |
[21] |
A. Jüngel and D. Matthes, Derivation of the isothermal quantum hydrodynamic equations using entropy minimization,, Z. Angew. Math. Mech., 85 (2005), 806.
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.
doi: 10.1137/050644823. |
[23] |
C. Levermore, Moment closure hierarchies for kinetic theories,, J. Stat. Phys., 83 (1996), 1021.
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. Google Scholar |
[25] |
F. Méhats and O. Pinaud, An inverse problem in quantum statistical physics,, J. Stat. Phys., 140 (2010).
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.
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.
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.
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.
doi: 10.1002/zamm.200900297. |
[5] |
A. Caldeira and A. Leggett, Path integral approach to quantum Brownian motion,, Physica A, 121 (1983), 587.
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.
|
[7] |
P. Degond, S. Gallego and F. Méhats, Isothermal quantum hydrodynamics: derivation, asymptotic analysis, and simulation,, Multiscale Model. Simul., 6 (2007), 246.
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, 1946 (2008), 111.
|
[9] |
P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum diffusion models derived from the entropy principle,, in, 12 (2006), 106.
|
[10] |
P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models,, J. Stat. Phys., 118 (2005), 625.
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.
doi: 10.1023/A:1023824008525. |
[12] |
J. Dong, A note on barotropic compressible quantum Navier-Stokes equations,, Nonlin. Anal., 73 (2010), 854.
doi: 10.1016/j.na.2010.03.047. |
[13] |
W. Dreyer, Maximisation of the entropy in non-equilibrium,, J. Phys. A, 20 (1987), 6505.
doi: 10.1088/0305-4470/20/18/047. |
[14] |
E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford Series in Mathematics and its Applications, 26 (2004).
|
[15] |
C. Gardner, The quantum hydrodynamic model for semiconductor devices,, SIAM J. Appl. Math., (1994), 409.
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.
doi: 10.1016/j.nonrwa.2010.11.005. |
[17] |
A. Jüngel, "Transport Equations for Semiconductors,", Lecture Notes in Physics, 773 (2009).
|
[18] |
A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids,, SIAM J. Math. Anal., 42 (2010), 1025.
|
[19] |
A. Jüngel, Effective velocity in compressible Navier-Stokes equations with third-order derivatives,, Nonlin. Anal., 74 (2011), 2813.
doi: 10.1016/j.na.2011.01.002. |
[20] |
A. Jüngel, Dissipative quantum fluid models,, to appear in Revista Mat. Univ. Parma, (2011). Google Scholar |
[21] |
A. Jüngel and D. Matthes, Derivation of the isothermal quantum hydrodynamic equations using entropy minimization,, Z. Angew. Math. Mech., 85 (2005), 806.
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.
doi: 10.1137/050644823. |
[23] |
C. Levermore, Moment closure hierarchies for kinetic theories,, J. Stat. Phys., 83 (1996), 1021.
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. Google Scholar |
[25] |
F. Méhats and O. Pinaud, An inverse problem in quantum statistical physics,, J. Stat. Phys., 140 (2010).
doi: 10.1007/s10955-010-0003-z. |
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