Existence and Asymptotic Behavior of Multi-Dimensional Quantum Hydrodynamic Model for Semiconductors
Existence and Asymptotic Behavior of Multi-Dimensional Quantum Hydrodynamic Model for Semiconductors
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DOI：
10.1007/s00220-003-1001-7
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发表时间：
2004-01
影响因子：
2.4
通讯作者：
Hai-liang Li;P. Marcati
中科院分区：
文献类型：
--
作者：
Hai-liang Li;P. Marcati
本文研究了空间周期域中关于电子粒子密度、电流密度和静电势的多维量子流体力学方程的存在性和时间渐近性。这些方程在形式上类似于经典流体力学，但动量方程不同，动量方程受到额外的非线性色散项（由于量子玻姆势）的影响，并用于半导体器件上量子效应的建模。在一般的、非凸的表达式-密度关系和大的正则初值条件下，我们证明了解的局部时间存在性。此外，我们提出了一个“亚音速”类型的稳定性条件有关的经典流体动力学方程。当满足这个条件时，局部时间解在时间上全局存在，并按时间指数收敛到相应的稳态。由于这个问题的经典方法，例如，对称双曲系统的弗里德里希理论不能使用，我们调查通过一个迭代过程的扩展系统，其中包括一个正在调查的特殊情况。特别是色散项出现在一个四阶波型方程的形式。
This paper is devoted to the study of the existence and the time-asymptotic of multi-dimensional quantum hydrodynamic equations for the electron particle density, the current density and the electrostatic potential in spatial periodic domain. The equations are formally analogous to classical hydrodynamics but differ in the momentum equation, which is forced by an additional nonlinear dispersion term, (due to the quantum Bohm potential) and are used in the modelling of quantum effects on semiconductor devices. We prove the local-in-time existence of the solutions, in the case of thegeneral, nonconvexpressure-density relation andlarge and regularinitial data. Furthermore we propose a ‘‘subsonic’’ type stability condition related to one of the classical hydrodynamical equations. When this condition is satisfied, the local-in-time solutions exist globally in-time and converge time exponentially toward the corresponding steady-state. Since for this problem classical methods like, for instance, the Friedrichs theory for symmetric hyperbolic systems cannot be used, we investigate via an iterative procedure an extended system, which incorporates the one under investigation as a special case. In particular the dispersive terms appear in the form of a fourth-order wave type equation.