粘性流体边界层理论的若干数学问题的研究

The strict proof of the boundary layer theory in Mathematics is still a challenge currently since it contains two difficult mathematical problems, i.e. the global well-posedness of the unsteady Prandtl boundary equation and the convergence of the boundary layer expansions. In recent years, studying the validity of the steady boundary layer expansions is a new tendency in this field, which is expected to find some new perspectives for the unsteady cases. In this direction, Prof. Yan Guo and the co-author contributed the pioneer paper, which established the validity of boundary layer expansions in a two-dimensional domain under the following assumptions: (1) horizontal length small enough, (2) boundary must move, (3)Euler flow is shear flow. Until now, there is still no reference obtaining a similar result with removing 2 or more conditions of the above 3. Therefore, establishing such validity for the boundary layer expansions without those constraints is the core goal of this project. In addition, as there have been several references on the invalidity of boundary layer expansions in some situations, this project will also explore the characteristics of some invalidity cases so that we are able to come up with some sufficient conditions to the validity in the future.

从数学上严格证明非稳态边界层理论是一个很有挑战性的问题，原因是非稳态Prandtl 边界层方程的整体适定性和边界层展开的收敛性是两个非常困难的数学问题。近来一个新的发展趋势是研究稳态Prandtl边界层展开的有效性，希望能为非稳态的问题寻找突破口。在这方面，布朗大学郭岩教授及其合作者首先进行了研究——在下面几个条件下证明了二维区域上的边界层展开的有效性：（1）水平宽度充分小；（2）边界必须运动；（3）欧拉流为剪切流。后来有些文献尝试放宽这些限制，但至今没有同时去除2个以上条件的成果。本项目的核心目标就是探讨新思路去除上述局限性条件证明边界层理论的有效性。另外，用稳定性分析方法探索非稳态边界层理论无效性的特征，以寻找保证边界层展开有效性的充分条件，是本项目的另一个研究重点。

从数学上严格证明非稳态边界层理论是一个很有挑战性的问题，原因是非稳态Prandtl边界层方程的整体适定性和边界层展开的收敛性是两个非常困难的数学问题。本项目进一步细分稳态Prandtl边界层的各层函数，通过延拓再光滑截断的策略，严格证明各层函数的存在性、正则性，最后得到收敛性，从而从一个侧面证明了Prandtl边界层理论在稳态层面的有效性。