Initial boundary value problems for a quantum hydrodynamic model of semiconductors: Asymptotic behaviors and classical limits

Nishibata, Shinya; Suzuki, Masahiro

doi:10.1016/j.jde.2007.10.035

Cited by 31 publications

(25 citation statements)

References 15 publications

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“…The standard continuation argument with Lemma 4.6 and Proposition 4.7 apparently yield the existence of the time global solution to the problem (1.4), (1.5), (1.9) and (1.10). The decay estimate (4.19) holds similarly as the arguments in [19] and [20]. 2…”

Section: Asymptotic Stability Of Stationary Solution With Small Datasupporting

confidence: 56%

“…For this model, the existence and the asymptotic stability of the stationary solution is also proven by the authors' previous paper [21]. In addition the isothermal hydrodynamic model with quantum correction is recently studied by the authors in [20], where the unique existence and the asymptotic stability of the stationary solution is also proven for the non-flat doping profile.…”

Section: Introductionmentioning

confidence: 71%

“…is derived in the similar way as in [19,20]. In this lemma, the existence time of the solution is denoted by T ε to make clear its dependence on ε.…”

Section: Asymptotic Behavior Of Solution To Drift-diffusion Modelmentioning

confidence: 99%

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Relaxation limit and initial layer to hydrodynamic models for semiconductors

Nishibata

Suzuki

2010

Journal of Differential Equations

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We study a relaxation limit of a solution to the initial-boundary value problem for a hydrodynamic model to a drift-diffusion model over a one-dimensional bounded domain. It is shown that the solution for the hydrodynamic model converges to that for the drift-diffusion model globally in time as a physical parameter, called a relaxation time, tends to zero. It is also shown that the solutions to the both models converge to the corresponding stationary solutions as time tends to infinity, respectively. Here, the initial data of electron density for the hydrodynamic model can be taken arbitrarily large in the suitable Sobolev space provided that the relaxation time is sufficiently small because the drift-diffusion model is a coupled system of a uniformly parabolic equation and the Poisson equation. Since the initial data for the hydrodynamic model is not necessarily in "momentum equilibrium", an initial layer should occur. However, it is shown that the layer decays exponentially fast as a time variable tends to infinity and/or the relaxation time tends to zero. These results are proven by the decay estimates of solutions, which are derived through energy methods.

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Section: Asymptotic Stability Of Stationary Solution With Small Datasupporting

confidence: 56%

Section: Introductionmentioning

confidence: 71%

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Relaxation limit and initial layer to hydrodynamic models for semiconductors

Nishibata

Suzuki

2010

Journal of Differential Equations

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“…For the transient QHD model, Jüngel et al [11] established the local existence of classical solutions. The global existence and large time behavior of smooth solutions for small initial data were obtained in [7,9,14,18]. Antonelli and Marcati [1] obtained the global existence of weak solutions to the QHD systems with arbitrarily large initial data in the energy norm.…”

Section: Introductionmentioning

confidence: 99%

Quasineutral limit of bipolar quantum hydrodynamic model for semiconductors

Yang

2011

Front. Math. China

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This paper is concerned with the quasineutral limit of the bipolar quantum hydrodynamic model for semiconductors. It is rigorously proved that the strong solutions of the bipolar quantum hydrodynamic model converge to the strong solution of the so-called quantum hydrodynamic equations as the Debye length goes to zero. Moreover, we obtain the convergence of the strong solutions of bipolar quantum hydrodynamic model to the strong solution of the compressible Euler equations with damping if both the Debye length and the Planck constant go to zero simultaneously.Keywords Bipolar quantum hydrodynamic model, quantum hydrodynamic equations, compressible Euler equations, quasineutral limit, modulated energy functional MSC 76Y05, 82D37, 82D10

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“…First results, e.g. [32,38,44], have been concerned with the local existence of solutions or the global existence of near-equilibrium solutions. For the stationary problem, only the existence of "subsonic" solutions has been achieved so far [29].…”

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confidence: 99%

Diffusive semiconductor moment equations using Fermi–Dirac statistics

Jüngel

Krause

Pietra

2010

Z. Angew. Math. Phys.

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Abstract. The global-in-time existence of weak solutions to the barotropic compressible quantum Navier-Stokes equations in a three-dimensional torus for large data is proved. The model consists of the mass conservation equation and a momentum balance equation, including a nonlinear thirdorder differential operator, with the quantum Bohm potential, and a density-dependent viscosity. The system has been derived by Brull and Méhats [10] from a Wigner equation using a moment method and a Chapman-Enskog expansion around the quantum equilibrium. The main idea of the existence analysis is to reformulate the quantum Navier-Stokes equations by means of a so-called effective velocity involving a density gradient, leading to a viscous quantum Euler system. The advantage of the new formulation is that there exists a new energy estimate which implies bounds on the second derivative of the particle density. The global existence of weak solutions to the viscous quantum Euler model is shown by using the Faedo-Galerkin method and weak compactness techniques. As a consequence, we deduce the existence of solutions to the quantum Navier-Stokes system if the viscosity constant is smaller than the scaled Planck constant.Key words. Compressible Navier-Stokes equations, quantum Bohm potential, density-dependent viscosity, global existence of solutions, viscous quantum hydrodynamic equations, third-order derivative, energy estimates. Gardner [20] from the Wigner equation by a moment method. More recently, dissipative quantum fluid models have been proposed. For instance, the moment method applied to the Wigner-Fokker-Planck equation leads to viscous quantum Euler models [21], and a Chapman-Enskog expansion in the Wigner equation leads under certain assumptions to quantum Navier-Stokes equations [10]. In this paper, we will reveal a connection between these two models by introducing an effective velocity variable, first used in capillary Korteweg-type models [5], and we will prove the global existence of weak solutions to the multidimensional initial-value problems for any finite-energy initial data. AMSIn the following, we describe the two dissipative quantum systems studied in this paper. The barotropic quantum Navier-Stokes equations for the particle density n and the particle velocity u read as

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Initial boundary value problems for a quantum hydrodynamic model of semiconductors: Asymptotic behaviors and classical limits

Cited by 31 publications

References 15 publications

Relaxation limit and initial layer to hydrodynamic models for semiconductors

Relaxation limit and initial layer to hydrodynamic models for semiconductors

Quasineutral limit of bipolar quantum hydrodynamic model for semiconductors

Diffusive semiconductor moment equations using Fermi–Dirac statistics

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