Asymptotic Stability of a Stationary Solution to a Thermal Hydrodynamic Model for Semiconductors

Nishibata, Shinya; Suzuki, Masahiro

doi:10.1007/s00205-008-0129-1

Cited by 56 publications

(52 citation statements)

References 17 publications

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“…The recent research by Guo and Strauss in [5] solved an important problem, the asymptotic stability of the stationary solution with the Dirichlet boundary condition for the general doping profile. Also, see [13].…”

Section: Related Resultsmentioning

confidence: 97%

“…In the following discussion, we follow the ideas in the paper [9,10] which have shown the existence of a time-local solution for the hyperbolic-elliptic coupled systems. Also, see [13]. …”

Section: An Existence Of a Solution Locally In Timementioning

confidence: 97%

“…The existence of the stationary solution to the hydrodynamic model is ensured in the next lemma (see [13] …”

Section: Introductionmentioning

confidence: 98%

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

Nishibata¹,

Suzuki²

2008

Journal of Differential Equations

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The present paper proves the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for a quantum hydrodynamic model of semiconductors over a one-dimensional bounded domain. We also discuss on a singular limit from this model to a classical hydrodynamic model without quantum effects. Precisely, we prove that a solution for the quantum model converges to that for the hydrodynamic model as the Planck constant tends to zero. Here we adopt a non-linear boundary condition which means quantum effect vanishes on the boundary. In the previous researches, the existence and the asymptotic stability of a stationary solution are proved under the assumption that a doping profile is flat, which makes the stationary solution also flat. However, the typical doping profile in actual devices does not satisfy this assumption. Thus, we prove the above theorems without this flatness assumption. Firstly, the existence of the stationary solution is proved by the Leray-Schauder fixed-point theorem. Secondly, we show the asymptotic stability theorem by using an elementary energy method, where the equation for an energy form plays an essential role. Finally, the classical limit is considered by using the energy method again.

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Section: Related Resultsmentioning

confidence: 97%

“…In the following discussion, we follow the ideas in the paper [9,10] which have shown the existence of a time-local solution for the hyperbolic-elliptic coupled systems. Also, see [13]. …”

Section: An Existence Of a Solution Locally In Timementioning

confidence: 97%

See 1 more Smart Citation

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

Nishibata¹,

Suzuki²

2008

Journal of Differential Equations

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show abstract

“…In the remainder of the introduction, we outline the motivation for this study. In [13], S. Nishibata and M. Suzuki characterized asymptotic limits of certain initial-boundary problems for the Euler-Poisson system. This paper followed the more comprehensive study of Y. Guo and W. Strauss [5], who also considered the multi-dimensional initial-boundary value problem and the associated exponential asymptotic convergence.…”

Section: Introductionmentioning

confidence: 99%

“…This paper followed the more comprehensive study of Y. Guo and W. Strauss [5], who also considered the multi-dimensional initial-boundary value problem and the associated exponential asymptotic convergence. The authors of [13] discuss in detail the steady problem, which is the focus of this paper. An earlier study was carried out by I. Gamba [4], who examined the steady isentropic model via an artificial viscosity method.…”

Section: Introductionmentioning

confidence: 99%

Steady Euler–Poisson systems: A differential/integral equation formulation with general constitutive relations

Jerome

2009

Nonlinear Analysis: Theory, Methods & Applications

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The Cauchy problem and the initial-boundary value problem for the Euler-Poisson system have been extensively investigated, together with a study of scaled and unscaled asymptotic limits. The pressure-density relationships employed have included both the adiabatic (isentropic) relation as well as the ideal gas law (isothermal). The study most closely connected to this one is that of S. Nishibata and M. Suzuki [Osaka J. Math. 44 (2007), 639-665], where a power law was employed in the context of the subsonic case, covering both the isothermal and adiabatic cases. These authors characterize the steady solution as an asymptotic limit. In this paper, we consider only the steady case, in much greater generality, and with more transparent arguments, than heretofore. We are able to identify both subsonic and supersonic regimes, and correlate them to one-sided boundary values of the momentum and concentration. We employ the novelty of a differential/integral equation formulation.

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Decay estimates of solutions to the bipolar compressible Euler–Poisson system in $$\pmb {\mathbb {R}^3}$$

Tong

Tan

2020

Z. Angew. Math. Phys.

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Asymptotic Stability of a Stationary Solution to a Thermal Hydrodynamic Model for Semiconductors

Cited by 56 publications

References 17 publications

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

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

Steady Euler–Poisson systems: A differential/integral equation formulation with general constitutive relations

Decay estimates of solutions to the bipolar compressible Euler–Poisson system in $$\pmb {\mathbb {R}^3}$$

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