Nonlinear structural stability and linear dynamic instability of transonic steady-states to hydrodynamic model for semiconductors
Nonlinear structural stability and linear dynamic instability of transonic steady-states to hydrodynamic model for semiconductors
复制标题
半导体流体动力学模型跨音速稳态的非线性结构稳定性和线性动态不稳定性
DOI：
10.1016/j.jde.2022.10.038
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发表时间：
2022-01
影响因子：
2.4
通讯作者：
Guojing Zhang
中科院分区：
文献类型：
--
作者：
Yue-Hong Feng;Ming Mei;Guojing Zhang
对于用Euler-Poisson方程表示的半导体器件的单极流体动力学模型，当掺杂浓度分布为超声速，边界数据分别位于亚音速区和超声速区时，系统具有激波跨音速定态和光滑跨音速定态。本文研究了这些定常跨音速解的非线性结构稳定性和线性动力不稳定性。对于任何松弛时间：0< τ≤+∞时，通过详细的奇异性分析，我们首先研究了当初始数据和掺杂分布的扰动足够小时，C1光滑跨音速定态的结构稳定性。我们注意到，当C1光滑跨音速定态通过音速线时，它们会产生系统的奇异性，从而给结构稳定性的证明带来一些本质上的困难。此外，当弛豫时间足够大时，在激波处电场为正的条件下，我们证明了跨音速激波稳态对于超声掺杂分布的小扰动是结构稳定的.此外，我们表明，线性动力学不稳定性，这些跨音速激波定常提供的电场是适当的负。结构稳定性结果的证明是基于奇异性分析、激波位置和下游密度的单调性论证以及超音速和亚音速解的稳定性分析。Euler-Poisson方程定常跨音速激波的线性动力学不稳定性可转化为Klein-Gordon方程自由边界问题的不适定性。利用非平凡变换和打靶法，证明了线性化问题存在指数增长的跨音速激波解。这些结果丰富和发展了现有的研究。
For unipolar hydrodynamic model of semiconductor device represented by Euler-Poisson equations, when the doping profile is supersonic, and the boundary data are in subsonic region and supersonic region separately, the system possesses the shock transonic steady-states and the smooth transonic steady-states. In this paper we study the nonlinear structural stability and the linear dynamic instability of these steady transonic solutions. For any relaxation time: 0< τ≤+∞, by means of elaborate singularity analysis, we first investigate the structural stability of the C 1-smooth transonic steady-states, once the perturbations of the initial data and the doping profiles are small enough. We note that, when the C 1-smooth transonic steady-states pass through the sonic line, they produce singularities for the system, and cause some essential difficulty in the proof of structural stability. Moreover, when the relaxation time is large enough τ≫ 1, under the condition that the electric field is positive at the shock location, we prove that the transonic shock steady-states are structurally stable with respect to small perturbations of the supersonic doping profile. Furthermore, we show the linearly dynamic instability for these transonic shock steady-states provided that the electric field is suitable negative. The proofs for the structural stability results are based on singularity analysis, a monotonicity argument on the shock position and the downstream density, and the stability analysis of supersonic and subsonic solutions. The linear dynamic instability of the steady transonic shock for Euler-Poisson equations can be transformed to the ill-posedness of a free boundary problem for the Klein-Gordon equation. By using a nontrivial transformation and the shooting method, we prove that the linearized problem has a transonic shock solution with exponential growths. These results enrich and develop the existing studies.
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DOI：
10.1137/20m1318869
发表时间：
2019-08
期刊：
SIAM J. Math. Anal.
影响因子：
--
作者：
S. Weng;Chunjing Xie;Z. Xin
通讯作者：
S. Weng;Chunjing Xie;Z. Xin
DOI：
10.1016/j.aml.2006.01.015
发表时间：
2006-12
期刊：
Appl. Math. Lett.
影响因子：
--
作者：
Yue-Jun Peng;Ingrid Violet
通讯作者：
Yue-Jun Peng;Ingrid Violet
DOI：
10.1007/978-3-7091-6961-2
发表时间：
1990
期刊：
--
影响因子：
--
作者：
P. Markowich;C. Ringhofer;C. Schmeiser
通讯作者：
P. Markowich;C. Ringhofer;C. Schmeiser
影响因子：
1
作者：
T. Luo;Z. Xin
通讯作者：
T. Luo;Z. Xin
DOI：
10.1007/bf00945711
发表时间：
1991-06
期刊：
Zeitschrift für angewandte Mathematik und Physik ZAMP
影响因子：
--
作者：
P. Markowich
通讯作者：
P. Markowich