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Thermal equilibrium solution to new model of bipolar hybrid quantum hydrodynamics. (English) Zbl 1368.35262
The paper presents system of equations for two liquids motivated by the problem of quasi-hydrodynamic description of electrons and holes in a semiconductor. For this reason, an additional term associated with the quantum effects (within Bohmian formalism) is added to each of basic partial stationary (by assumption of the thermal equilibriom) differential equations. Further, the system is treated purely mathematically, and an existence of limit functions, which are the weak solutions to the system considered, is proven.

MSC:
35Q82 PDEs in connection with statistical mechanics
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
82D37 Statistical mechanical studies of semiconductors
35D30 Weak solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
Software:
COLNEW; COLSYS
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[1] Ancona, M. G.; Iafrate, G. J., Quantum correction to the equation of state of an electron gas in a semiconductor, Phys. Rev. B, 39, 9536-9540, (1989)
[2] Antonelli, P.; Marcati, P., The quantum hydrodynamics system in two space dimensions, Arch. Ration. Mech. Anal., 203, 499-527, (2012) · Zbl 1290.76165
[3] Ascher, U.; Christiansen, J.; Russell, R., Collocation software for boundary-value odes, ACM Trans. Math. Software, 7.2, 209-222, (1981) · Zbl 0455.65067
[4] Bader, G.; Ascher, U., A new basis implementation for a mixed order boundary value ODE solver, SIAM J. Sci. Statist. Comput., 8, 483-500, (1987) · Zbl 0633.65084
[5] Baro, M.; Ben Abdallah, N.; Degond, P.; El Ayyadi, A., A 1D coupled Schrödinger drift-diffusion model including collisions, J. Comput. Phys., 203, 129-153, (2005) · Zbl 1067.82060
[6] Ben Abdallah, N.; Pietra Jourdana, P.; Vauchelet, N., A hybrid classical-quantum approach for ultra-scaled confined nanostructures: modeling and simulation, (ESAIM: Proceedings, vol. 35, (2012), EDP Sciences), 239-244 · Zbl 1357.81094
[7] Ben Abdallah, N., A hybrid kinetic-quantum model for stationary electron transport, J. Stat. Phys., 90, 627-662, (1998) · Zbl 0949.76075
[8] Brezzi, F.; Gasser, I.; Markowich, P. A.; Schmeiser, C., Thermal equilibrium states of the quantum hydrodynamic model for semiconductors in one dimension, Appl. Math. Lett., 8, 47-52, (1995) · Zbl 0829.35127
[9] Chen, Q.; Guan, P., Weak solutions to the stationary quantum drift-diffusion model, J. Math. Anal. Appl., 359, 666-673, (2009) · Zbl 1173.35644
[10] Chiarelli, S.; di Michele, F.; Rubino, B., A hybrid drift diffusion model: derivation, weak steady state solutions and simulations, Math. Appl., 1, 37-55, (2012) · Zbl 1312.35165
[11] Degond, P.; Markowich, P. A., On a one-dimensional steady-state hydrodynamic model for semiconductors, Appl. Math. Lett., 3, 25-29, (1990) · Zbl 0736.35129
[12] Di Michele, F.; Marcati, P.; Rubino, B., Steady states and interface transmission conditions for heterogeneous quantum classical 1-d hydrodynamic model of semiconductor devices, Phys. D, 243, 1-13, (2013) · Zbl 1271.82029
[13] Di Michele, F.; Marcati, P.; Rubino, B., Stationary solution for transient quantum hydrodynamics with Bohmenian-type boundary conditions, Comput. Appl. Math., 36, 459-479, (2017) · Zbl 1457.76206
[14] Di Michele, F.; Mei, M.; Rubino, B.; Sampalmieri, S., Solutions to hybrid quantum hydrodynamical model of semiconductors in bounded domain, Int. J. Numer. Anal. Model., 13, 898-925, (2016) · Zbl 1362.35246
[15] Di Michele, F.; Rubino, B.; Sampalmieri, S., A steady-state mathematical model for an EOS capacitor: the effect of the size exclusion, Netw. Heterog. Media, 11, 603-625, (2016) · Zbl 1354.82028
[16] Gardner, C. L., The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54, 409-427, (1994) · Zbl 0815.35111
[17] Guo, Y.; Strauss, W., Stability of semiconductor states with insulating and contact boundary conditions, Arch. Ration. Mech. Anal., 179, 1-30, (2005) · Zbl 1148.82030
[18] Gyi, M. T.; Jüngel, A., A quantum regularization of the one-dimensional hydrodynamic model for semiconductors, Adv. Differential Equations, 5, 773-800, (2000) · Zbl 1174.82348
[19] Huang, F.; Mei, M.; Wang, Y.; Yu, H., Asymptotic convergence to stationary waves for unipolar hydrodynamic model of semiconductors, SIAM J. Math. Anal., 43, 411-429, (2011) · Zbl 1227.35063
[20] Jourdana, C.; Pietra, P., A hybrid classical-quantum transport model for the simulation of carbon nanotube transistors, SIAM J. Sci. Comput., 36, B486-B507, (2014) · Zbl 1306.82021
[21] Jüngel, A.; Li, H., On a one-dimensional steady-state hydrodynamic model, Arch. Math. (Brno), 40, 435-456, (2004)
[22] Jüngel, A.; Li, H., Quantum Euler-Poisson systems: global existence and exponential decay, Quart. Appl. Math., 62, 569-600, (2004) · Zbl 1069.35012
[23] Li, H.; Marcati, P., Existence and asymptotic behavior of multi-dimensional quantum hydrodynamic model for semiconductors, Comm. Math. Phys., 245, 215-247, (2004) · Zbl 1075.82019
[24] Li, H.; Markowich, P. A.; Mei, M., Asymptotic behaviour of solutions of the hydrodynamic model of semiconductors, Proc. Roy. Soc. Edinburgh, 132A, 359-378, (2002) · Zbl 1119.35310
[25] Luo, T.; Natalini, P.; Xin, Z., Large time behavior of the solutions to a hydrodynamic model for semiconductors, SIAM J. Appl. Math., 59, 810-830, (1998) · Zbl 0936.35111
[26] Marcati, P.; Natalini, R., Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Arch. Ration. Mech. Anal., 129, 129-145, (1995) · Zbl 0829.35128
[27] Nishibata, S.; Suzuki, M., Initial boundary value problems for a quantum hydrodynamic model of seminconductors: asymptotic behaviors and classical limits, J. Differential Equations, 244, 836-874, (2008) · Zbl 1139.82042
[28] Pacard, F.; Unterreiter, A., A variational analysis of the thermal equilibrium state of charged quantum fluids, Comm. Partial Differential Equations, 20, 885-900, (1995) · Zbl 0820.35112
[29] Salas, O.; Lanucara, P.; Pietra, P.; Rovida, S.; Sacchi, G., Parallelization of a quantum-classic hybrid model for nanoscale semiconductor devices, Rev. Mat. Teor. Apl., 18, 231-248, (2011) · Zbl 1307.82027
[30] Unterreiter, A., The thermal equilibrium solution of a generic bipolar quantum hydrodynamic model, Comm. Math. Phys., 188, 69-88, (1997) · Zbl 1004.82507
[31] Zhang, G.; Li, H.-L.; Zhang, K., Semiclassical and relaxation limits of bipolar quantum hydrodynamic model for semiconductors, J. Differential Equations, 245, 1433-1453, (2008) · Zbl 1154.35071
[32] Zhang, G.; Zhang, K., On the bipolar multidimensional quantum Euler-Poisson system: the thermal equilibrium solution and semiclassical limit, Nonlinear Anal., 66, 2218-2229, (2007) · Zbl 1151.82373
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