Title: Quasisymmetric magnetic fields in asymmetric toroidal domains Show Chinese title

Title: 非对称环形区域中的准对称磁场

Abstract: We explore the existence of quasisymmetric magnetic fields in asymmetric toroidal domains. These vector fields can be identified with a class of magnetohydrodynamic equilibria in the presence of pressure anisotropy. First, using Clebsch potentials, we derive a system of two coupled nonlinear first order partial differential equations expressing a family of quasisymmetric magnetic fields in bounded domains. In regions where flux surfaces and surfaces of constant field strength are not tangential, this system can be further reduced to a single degenerate nonlinear second order partial differential equation with externally assigned initial data. Then, we exhibit regular quasisymmetric vector fields which correspond to local solutions of anisotropic magnetohydrodynamics in asymmetric toroidal domains such that tangential boundary conditions are fulfilled on a portion of the bounding surface. The problems of boundary shape and locality are also discussed. We find that symmetric magnetic fields can be fitted into asymmetric domains, and that the mathematical difficulty encountered in the derivation of global quasisymmetric magnetic fields lies in the topological obstruction toward global extension affecting local solutions of the governing nonlinear first order partial differential equations. Show Chinese abstract

Abstract: 我们探讨了在非对称环形区域中准对称磁场的存在性。 这些矢量场可以与存在压力各向异性的磁流体动力学平衡的一类情况相联系。 首先，利用Clebsch势，我们推导出一组两个耦合的非线性一阶偏微分方程，用于表达有限区域内一族准对称磁场。 在通量面和场强恒定面不相切的区域，该系统可以进一步简化为一个带有外部指定初始数据的退化非线性二阶偏微分方程。 然后，我们展示出规则的准对称矢量场，这些场对应于非对称环形区域中各向异性磁流体动力学的局部解，并且在边界表面的一部分上满足切向边界条件。 边界形状和局部性问题也进行了讨论。 我们发现对称磁场可以适配到非对称域中，并且在推导全局准对称磁场时遇到的数学困难在于影响控制非线性一阶偏微分方程局部解的全局扩展的拓扑障碍。