MHD方程边界层问题的研究

边界层理论是流体力学中的一个重要课题。大量的实验数据和数值结果都支持这一理论。然而，研究边界层理论的数学基础问题却非常困难，至今已取得的研究成果也很少。本申请项目拟研究MHD方程的边界层问题。MHD方程描述了导电流体在电磁场中的运动状态，在天体物理学、地球物理、空气动力学等领域具有广泛的应用，是流体力学中最重要的模型之一。本申请项目拟研究：(1)一维可压缩MHD方程的边界层效应和边界层厚度估计，包括三个方面：一是当粘性系数趋于零时，边界层厚度函数与粘性系数之间的关系；二是当磁扩散系数趋于零时，边界层厚度与磁扩散系数之间的关系；三是当粘性系数和磁扩散系数都趋于零时(例如两个系数相等且趋于零)，边界层厚度与两个系数之间的关系。考虑的状态方程包括两类情形：一类是理想气体情形，另一种是受热辐射影响的情形。(2)具非特征边界的三维不可压缩MHD方程的边界层。在研究过程中，需要克服由磁场的影响。

The boundary layer theory is an important subject in fluid mechanics. A large number of experimental data and numerical results support the theory. However, it is very difficult to study problem on mathematical basis of boundary layer theory, so far there are few results. In this project, we mainly study problems on boundary layer for MHD equations. MHD, which concerns the motion of conducting fluids in an electromagnetic field, has a very broad range of applications in many fields, such as astrophysics, geophysics and aerodynamics, and is one of the most important models in fluid mechanics. The main interest of this project is twofold. First, we will study the boundary layer effect and boundary layer thickness for 1D compressible MHD equations, which include three problems: the relation between the thickness function and the viscosity coefficient as the viscosity coefficient tends to zero, the relation between the thickness function and the magnetic diffusion coefficient as the diffusion coefficient tends to zero, and the relation among the thickness function, the viscosity coefficient and the diffusion coefficient as the two coefficients tend to zero at the same time (for example, they are equal and tend to zero). The state equations we consider here include both the ideal gas case and the case of thermal radiation. Moreover, we will study boundary layer for 3D incompressible MHD equation with non-characteristic boundary. For these, we need to overcome some serious difficulties due to the presence of the magnetic field and its complex interaction with the hydrodynamic motion.