磁流体力学 (Magnetohydrodynamics) 方程, 一般简称为 MHD 方程, 它描述了等离子体和磁场的相互作用, 由经典的流体力学中的 Navier-Stokes 方程和电动力学中的 Maxwell 方程耦合而成. 本文主要讨论一类稳态不可压 MHD 方程的边值问题, 研究了在多连通有界区域的情形下, 边界上的流量满足不同条件时, 相应的非齐次边值问题解的存在性, 主要利用了椭圆方程的弱解理论, 不动点定理, Galerkin 逼近, Morse-Sard 定理等.
本文共分为四章:
第一章主要介绍了稳态不可压 MHD 方程模型及其研究现状, 给出了本文所需的预备知识并简要阐述了本文的主要结果.
第二章讨论了, 在多连通有界域上, 方程的边值满足强制条件 (每个边界分支上的流量均为 0) 时解的存在性.  应用反证法, 并利用椭圆方程的弱解理论和 Leray-Schauder 不动点定理, 通过引进矩阵变量, 将~MHD~方程统一的写成矩阵方程, 进而定义映射, 构造出假设的矛盾, 从而给出存在性结论.
第三章讨论了, 在多连通有界域上, 将第二章中的强制条件减弱, 即在只满足边界流量足够小的条件时方程解的存在性. 首先使用 Sobolev 延拓定理将非齐次边值问题转化为齐次边值问题, 再由边界流量满足小性条件, 利用 Galerkin 方法得到了解的存在性.  
第四章讨论了, 在多连通有界域上, 边界流量仅满足相容条件时解的存在性. 首先应用反证法假设条件和 Leray-Schauder 不动点定理找到一个无界的函数列, 然后构造了一个新的能量函数, 并利用 Morse-Sard 定理在流函数的一些水平线上得到这个能量函数梯度的一致估计. 最后结合 Coarea 公式得到这些水平线的 Hausdorff 测度的一组相矛盾的上下界估计, 进而证明了方程解的存在性. 由此证明过程可知, 磁场在流体中起到了稳定性的作用.

Magnetohydrodynamics, or MHD system for short, describes the interaction of plasma and magnetic field. It is a combination of the Navier-Stokes equations of fluid dynamics Maxwell's equations of electromagnetism. In this paper, we mainly discuss the boundary value problems of a kind of steady incompressible MHD system. We study the existence of the solution of the nonhomogeneous boundary value problems when the boundary of the domain has more than one connected component and the flux on each boundary connected component satisfies different conditions using the weak solution theory of elliptic equations, fixed point theorem, Galerkin's method and Morse-Sard theorem, etc.
The thesis consists of four chapters:
In Chapter 1, the basic concept of the steady incompressible MHD equations model and the recent development are introduced, as well as some preliminary knowledge which will be used in the paper, and the main results of the dissertation are outlined.
In Chapter 2, we study the existence of solutions of the equation in a multiconnected bounded domain when the boundary value satisfies the mandatory condition that the flow on each boundary branch is 0. By the method of reduction to absurdity. Using the weak solution theory of elliptic equation and Leray Schauder fixed point theorem. By introducing matrix variables, we can write MHD equations as a matrix equation and define a mapping. Then we construct a contradiction with the hypothesis and proof the conclusion.
In Chapter 3, we study the existence of the solutions of the equation when the constraint condition in Chapter 2 is weakened, namely the boundary flow is small enough. We first use the Sobolev extension method to transform the nonhomogeneous boundary value problem into homogeneous boundary value problem. Since the boundary flow satisfies the small condition, we can obtain the existence of the solutions by using Galerkin's method.
In Chapter 4, we study the existence of solutions in multiconnected bounded domains when the boundary flow only satisfies the compatibility condition. First using the method of reduction to absurdity and the Leray Schauder fixed point theorem, we can find an unbounded sequence of functions. Next we construct an energy function, and we get a uniform estimate of the gradient of the energy function on some level lines of stream function by the Bernoulli theorem. Combining with the Coarea formula, we get a set of contradictory upper and lower bound of Hausdorff measures of the level lines, which proves the existence of the solutions. From the proof, we find that the magnetic field plays a role of stability in the fluid.