论文标题
非线性结构稳定性和跨气体稳态与半导体流体动力模型的线性动态不稳定性
Nonlinear structural stability and linear dynamic instability of transonic steady-states to a hydrodynamic model for semiconductors
论文作者
论文摘要
对于由Euler-Poisson方程表示的半导体装置的单极流体动力学模型，当掺杂曲线是超音速时，分别在[27]和[41]中分别建立了稳定的透射冲击溶液和Euler-Poisson方程的C稳定稳定的跨音速溶液。在本文中，我们进一步研究了这些稳定的跨音速溶液的非线性结构稳定性和线性动态不稳定性。当C^1平滑的跨音速稳态穿过声音线时，它们会为系统产生奇异性，并在结构稳定性的证明方面造成了一些重要的困难。在任何放松时间，通过精细的奇异性分析，一旦初始数据的扰动和掺杂曲线的扰动足够小，我们首先研究了C^1平滑的跨音速稳态的结构稳定性。此外，当放松时间足够大时，在电场位置在电场位置为正时，我们证明跨性别冲击稳态相对于超音速掺杂曲线的小扰动在结构上是稳定的。此外，我们显示了这些跨性能冲击稳态的线性动态不稳定性，前提是电场合适。结构稳定性结果的证明是基于奇异性分析，关于冲击位置和下游密度的单调论点以及超音速和亚音速溶液的稳定性分析。 Euler-Poisson方程的稳定透射冲击的线性动态不稳定性可以转化为Klein-Gordon方程的自由边界问题的不良性。通过使用非平凡的转换和拍摄方法，我们证明了线性化问题具有带指数增长的透射冲击解决方案。这些结果丰富了现有的研究。
For unipolar hydrodynamic model of semiconductor device represented by Euler-Poisson equations, when the doping profile is supersonic, the existence of steady transonic shock solutions and C-smooth steady transonic solutions for Euler-Poisson Equations were established in [27] and [41], respectively. In this paper we further study the nonlinear structural stability and the linear dynamic instability of these steady transonic solutions. 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. For any relaxation time, 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. Moreover, when the relaxation time is large enough, 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.
