磁流体动力学方程边界层数学理论中的若干问题研究

磁流体动力学(MHD)方程能够用来描述运动的导电流体和磁场间的相互作用，由于其应用的广泛性和结构的复杂性而成为国际数学界的热点领域之一。特别当出现边界层效应时，MHD方程的数学分析更为困难，至今尚有不少理论空白。本项目旨在研究MHD方程边界层理论中亟待解决的几个问题，期望通过观察MHD方程及相应边界层方程自身结构所蕴含的本质结构机制发展一些新的分析技巧。主要围绕两个问题展开探讨：(1)二维不可压无磁扩散MHD方程在无滑移边界条件下的高雷诺数极限。致力于严格证明该方程边界层数学理论的合理性。(2)二维MHD边界层方程的适定性。致力于在没有任何结构性条件的前提下建立该方程的适定性理论。 . 这些问题不仅有着重要的实际应用背景，而且具有很高的数学挑战度。我们希望在这些问题的研究上取得进展，一方面能够丰富MHD方程边界层的数学理论，另一方面有助于加深对磁场效应的认识。

Magnetohydrodynamics (MHD) equations can be used to describe the interaction between the moving conductive fluids and the magnetic field. Because of its wide applications and complex structures, it has become one of the hot fields in the international mathematics field. In particular, the mathematical analysis of MHD system is more difficult when the boundary layer effect appears. This project aims to investigate several problems on the boundary layer effect on the MHD system and develop some new analytic techniques by observing the essential structural mechanisms in both the MHD system and the corresponding boundary layer systems. We will focus on the following two problems: (1) The small viscosity limit of the 2D incompressible MHD system without resistivity under the no-slip boundary condition on the velocity. The main purpose is to justify the validity of the boundary layer theories of the system. (2) Well-posedness of solutions for the 2D MHD boundary layer equations. One of the main purpose is to show the well-posedness of solutions for the system without any Structure Assumption.. These problems not only have important practical background, but also contain the challenging mathematical difficulties. We hope to make some progresses on the problems above, which will enrich the boundary layer theories and deepen the understanding of magnetic effect.