几类半导体流体动力学模型的定性研究

In this project, we plan to use modern theoretical tools of partial differential equation (PDE) to investigate the mathematical properties of several types of semiconductor hydrodynamic models. The concerned models cover the two levels of the hydrodynamic model (Euler-Poisson system) and the quantum hydrodynamic model (quantum Euler-Poisson system), and their mathematical structures are very complicated. The key issues studied are the following: the well-posedness and profile analysis of steady-state solutions, the local and global well-posedness of transient solutions (In one-dimensional case, we study the global classical solutions with large data; In multi-dimensional case, we explore the global classical solutions with small data and the global weak solution with large data), the large-time behavior (asymptotic stability and nonlinear diffusion phenomenon), and the singular limits (zero relaxation-time limit, semi-classical limit and quasi-neutral limit). Concerned about how the multiple factors of the doping profile, boundary effect, heat exchange, quantum effect and recombination-generation affect the models. In particular, we are going to establish the mathematical theory of a full quantum hydrodynamic (FQHD) model which takes the energy equation with the third-order dispersive velocity term into account. Until now, there are only a few numerical results on the FQHD model. These mathematical problems belong to the international frontier topics, and have a broad application prospect. The related expected results have the important significance of developing the theory of semiconductor PDEs.

本项目拟用现代偏微分方程的理论工具研究几类半导体流体动力学模型的数学性质。关注的模型涵盖流体动力学模型（Euler-Poisson方程组）和量子流体动力学模型（量子Euler-Poisson方程组）两个层面，且具有复杂的数学结构。重点研究的问题涉及稳态解的适定性和结构分析、瞬态解的局部适定性和整体适定性（一维模型讨论大初值整体古典解；高维模型探讨小初值整体古典解和大初值整体弱解）、解的大时间行为（渐近稳定性和非线性扩散现象）和解的奇异极限（零松弛时间极限、半经典极限和拟中性极限）。关心掺杂分布、边界效应、热交换、量子效应和重组增生等多重因素对模型的影响。特别地，本项目研究含有带三阶色散速度项能量方程在内的完整量子流体动力学（FQHD）模型的数学理论，目前国际上针对FQHD模型只有少部分数值结果。这些问题是国际前沿课题，具有广阔应用前景。相关预期成果具有完善半导体偏微分方程理论的重要意义。

本项目主要研究几类半导体宏观流体动力学模型的数学性质，取得了若干重要进展。（1）我们研究了一维稳态双极FQHD模型的边值问题，得到了亚音速稳态解的存在唯一性，建立了解关于多尺度参数的一致估计，验证了稳态模型间的层级关系；（2）我们研究了一维单极FQHD模型的初边值问题，在非平坦掺杂分布假设下，得到了亚音速稳态解的存在唯一性，瞬态解的局部适定性、小初值整体适定性和大时间行为，验证了稳态和瞬态解的半经典极限；（3）我们研究了三维带重组增生率的双极FHD模型的初边值问题，在绝缘边值条件下，我们得到了非常数热平衡解的存在唯一性和指数渐近稳定性；（4）我们还研究了一维单极等温QHD模型初边值问题大初值整体适定性、三维双极FQHD模型周期问题解的整体适定性和大时间行为、一维双极FQHD模型Cauchy问题的非线性扩散现象。这些结果中，已有2篇文章发表在SCI期刊《Applicable Analysis》上，1篇文章已投稿，其余结果正在整理成论文形式。