两类流体动力学模型的渐近机理和边界层问题

Asymptotic analysis and boundary layer problems of the hydrodynamics models coupled with multi-physical fields will be studied in the program. The special studies are the quasineutral limits and boundary layer, initial layer, mixed layer problems, which include: the quasineutral limit and boundary layer problem of the multi-particle drift-diffusion model in half space of three dimension, the quasineutral limit and boundary layer, mixed layer problems of electro-diffusion arising in electrohydrodynamics in a bounded domain in three dimensional space. .The asymptotic limits of the models with small parameters could reveal the asymptotic relationship between the complex fluid coupled with multi-physical field and the classical fluid, and explicate the phenomenon of numerical simulation. It is better and helpful for human to understand the nonlinear phenomenon of the natural environment. On mathematical theory, the stability of the multi-scaling structure and boundary theory could be obtained from the studies of the asymptotic limits of the complex fluid coupled with multi-physical field..The models in this program have been applied extensively in modern chemistry, biology, pharmacology, environmental science, material science and semiconductor industry. The asymptotic analysis and boundary layer problems are of challenge and frontier in the field of nonlinear partial differential equations, and have vital significance in both mathematical theory and the practical applications.

本项目研究两类流体动力学模型的渐近机理和边界层问题，重点研究半导体漂流扩散（drift-diffusion）模型、电解液电扩散模型的拟中性极限和边界层、初始层、混合层问题，包括半空间中多粒子漂流扩散模型(扩散一致和非一致时)的拟中性极限和边界层问题和三维有界区域中电解液电扩散模型（扩散一致和非一致时）的拟中性极限和边界层、混合层问题。.小参数的渐近极限问题揭示了多物理场耦合下复杂模型与经典流体动力学模型之间的渐近关系，解释了数值模拟现象和自然环境中的非线性现象；数学上，对模型渐近机理的研究可以获得多物理场耦合下流体动力学模型的多尺度结构稳定性和边界层理论。.本项目研究的两类模型在现代化学、生物学、药物学、环境科学、材料学和半导体工业界有广泛应用，对模型的渐近机理和边界层问题的研究是偏微分方程领域极具挑战的前沿课题，在数学理论和应用科学上都具有十分重要的意义。

本项目按照计划研究了应用科学领域中流体动力学模型的渐近机理和边界层问题，项目组成员与合作者研究了一维空间当电子和正电荷迁移率不相等时半导体漂流扩散模型的拟中性极限和混合层问题；三维空间中电子和正电荷扩散系数相同和不相同时电解液电扩散模型的拟中性极限和混合层问题；多粒子扩散模型的拟中性极限问题。多物理场耦合的流体动力学模型在解决现实科学问题中起重要作用。小参数的渐近极限问题揭示了多物理场耦合下复杂模型与经典流体动力学模型之间的渐近关系，解释了数值模拟现象和自然环境中的非线性现象，在数学理论上具有挑战性。这些研究内容不仅是国内外数学家、物理学家和生物学家等广泛关注的研究热点，而且还是应用科学和工程技术领域的研究课题。项目针对以上内容开展相关问题的研究，在数学理论方面取得研究成果有：在SCI收录期刊已发表学术论文1篇，投稿论文2篇，完成本项目的预期目标。