In this paper, we continue to analyse the properties of the hybrid model introduced in the works by Chiarelli et al. and Di Michele et al. In particular, we consider a quantum hybrid model with a special viscous functional, as introduced in the work by Gamba and Jüngel, for the quantum hydrodynamical model in its standard form (QHD). We show that the hybrid QHD (H-QHD) model admits weak solutions also for arbitrary large value of the current density J under a suitable pressure function.
Ben AbdallahNJourdanaCPietraP, et al. A hybrid classical-quantum approach for ultra-scaled confined nanostructures: modeling and simulation. ESAIM: Procs2012; 35: 239–244.
2.
Ben AbdallahN. A hybrid kinetic-quantum model for stationary electron transport. J Statistical Physics1998; 90: 627–662.
3.
BaroMBen AbdallahNDegondP, et al. A 1D coupled Schrodinger drift-diffusion model including collisions. J Comput Phys2005; 203: 129–153.
4.
Di MicheleFMarcatiPRubinoB. Steady states and interface transmission conditions for heterogeneous quantum classical 1-d hydrodynamic model of semiconductor devices. Physica D2013; 243: 1–13.
5.
Di MicheleFMarcatiPRubinoB. Stationary solution for transient quantum hydrodynamics with bohmenian-type boundary conditions. Comput Appl Math2017; 36(1): 459–479.
6.
SalasOLanucaraPPietraP, et al. Parallelization of a quantum-classic hybrid model for nanoscale semiconductor devices. Revista Matem Teor Apli2011; 18: 231–248.
7.
JourdanaCPietraP. A hybrid classical-quantum transport model for the simulation of carbon nanotube transistors. SIAM J Sci Comput2014; 36: B486–B507.
8.
Di MicheleFMeiMRubinoB, et al. Stationary solutions to hybrid quantum hydrodynamical model of semiconductor in bounded domains. Int J Num Anal Model2016; 13: 898–925.
9.
Di MicheleFMeiMRubinoB, et al. Thermal equilibrium solution to new model of bipolar hybrid quantum hydrodynamics. J Differ Equ2017; 263(3): 1843–1873.
10.
Di MicheleFMeiMRubinoB, et al. Stationary solutions for a new hybrid quantum model for semiconducotrs with discontinuous pressure functional and relaxation time. Math Mech Solids2019; 24: 2096–2115.
11.
Di MicheleFMeiMRubinoB, et al. Existence and uniqueness for a stationary hybrid quantum hydrodynamical model with general pressure functional. Commun Math Sci2021; 19(8): 2049–2079.
12.
HaasF. A magnetohydrodynamic model for quantum plasmas. Phys Plasmas2005; 12(6): 0621171.
13.
VasseurAFYuC. Global weak solutions to the compressible quantum Navier–Stokes equations with damping. SIAM J Math Anal2016; 48(2): 1489–1511.
14.
Di MicheleFRubinoBSampalmieriR. A steady-state mathematical model for an EOS capacitor: the effect of the size exclusion. Netwo Heterogen Media2016; 11(4): 603–625.
15.
AntonelliPMarcatiP. The quantum hydrodynamics system in two space dimensions. Arch Ration Mech2012; 203: 499–527.
16.
GyiAMTJüngelA. A quantum regularization of the one-dimensional hydrodynamic model for semiconductors. Adv Diff Equ2000; 5: 773–800.
17.
JüngelALiH. On a one-dimensional steady-state hydrodynamic model. Arch Math2004; 40: 435–456.
18.
JüngelALiH. Quantum Euler-Poisson systems: global existence and exponential decay. Quart Appl Math2004; 62: 569–600.
19.
NishibataSSuzukiM. Initial boundary value problems for a quantum hydrodynamic model of semiconductors: asymptotic behaviors and classical limits. J Differ Equ2008; 244: 836–874.
20.
ChiarelliSDi MicheleFRubinoB. A hybrid drift diffusion model: derivation, weak steady state solutions and simulations. Math Appl2012; 1: 37–55.
21.
ChenLDreherM. The viscous model of quantum hydrodynamics in several dimensions. Math Model Method Appl Sci2007; 17(7): 1065–1093.
22.
JüngelAMilišićJP. Physical and numerical viscosity for quantum hydrodynamics. Commun Math Sci2007; 5(2): 447–471.
23.
DiawAMurilloMS. A viscous quantum hydrodynamics model based on dynamic density functional theory. Sci Rep2017; 7(1): 1–9.
24.
DreherM. The transient equations of viscous quantum hydrodynamics. Math Method Appl Sci2008; 31(4): 391–414.
25.
GambaIMJüngelA. Positive solutions to singular second and third order differential equations for quantum fluids. Arch Ration Mech2001; 156: 183–203.
26.
GambaIM. Stationary transonic solution of a one-dimensional hydrodynamic model for semiconductor. Commun Part Differ Equ1992; 17(3–4): 225–267.
27.
FangWKazufumiI. Steady-state solutions of a one-dimensional hydrodynamic model for semiconductors. J Differ Equ1997; 133(2): 224–244.
28.
BallestraLMichelettiSSaccoR, et al. On a viscous-hydrodynamic model for semiconductors: numerical simulation and stability analysis. Comput Visual Sci2001; 4(2): 79–86.
29.
AnconaMGIafrateGJ. Quantum correction to the equation of state of an electron gas in a semiconductor. Phys Rev B1989; 39: 9536–9540.
30.
BrezziFGasserIMarkowichPA, et al. Thermal equilibrium states of the quantum hydrodynamic model for semiconductors in one dimension. Appl Math Lett1995; 8: 47–52.
31.
DegondPMarkowichPA. On a one-dimensional steady-state hydrodynamic model for semiconductors. Appl Math Lett1990; 3: 25–29.
32.
GardnerCL. The quantum hydrodynamic model for semiconductor devices. SIAM J Appl Math1994; 54: 409–427.
33.
GuoYStraussW. Stability of semiconductor states with insulating and contact boundary conditions. Arch Ration Mech2005; 179: 1–30.
34.
HuangFMeiMWangY, et al. Asymptotic convergence to stationary waves for unipolar hydrodynamic model of semiconductors. SIAM J Math Anal2011; 43: 411–429.
35.
LiHMarkowichPAMeiM. Asymptotic behaviour of solutions of the hydrodynamic model of semiconductors. Proc Royal Soc Edinburgh2002; 132A: 359–378.
36.
LuoTNataliniPXinZ. Large time behavior of the solutions to a hydrodynamic model for semiconductors. SIAM J Appl Math1998; 59: 810–830.
37.
MarcatiPNataliniR. Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation. Arch Ration Mech1995; 129: 129–145.