非简并环形区域中具有非定常压力的MHD平衡
MHD equilibria with nonconstant pressure in nondegenerate toroidal domains
摘要：
本文证明了在三维环形$\mathbf{R}^3$区域中存在分段光滑的MHD平衡点，其中边界上的压力是常数，而内部不是。压力是分段恒定的，等离子体电流表现出任意数量的电流片。我们还建立了存在的自由边界稳定状态所包围的真空与外部表面电流。这些平衡被证明存在的环形域不需要是轴对称域的小扰动，事实上它们可以有任何打结拓扑。我们在构造中使用的积木是满足一定的非简并条件的解析环形域，该条件大致说明在域的表面上存在遍历的无力场。证明涉及三个主要成分：一个分段光滑MHD平衡点的胶合结构，二维环面上的Hamilton-Jacobi方程，可以理解为上同调方程的非线性变形（因此非简并假设在相应的分析中起着重要作用），和一个新的KAM定理，专门用于研究三维无发散场，其庞加莱映射不能显式计算。
英文摘要：
We prove the existence of piecewise smooth MHD equilibria in three-dimensional toroidal domains of $\mathbf{R}^3$ where the pressure is constant on the boundary but not in the interior. The pressure is piecewise constant and the plasma current exhibits an arbitrary number of current sheets. We also establish the existence of free boundary steady states surrounded by vacuum with an external surface current. The toroidal domains where these equilibria are shown to exist do not need to be small perturbations of an axisymmetric domain, and in fact they can have any knotted topology. The building blocks we use in our construction are analytic toroidal domains satisfying a certain nondegeneracy condition, which roughly states that there exists a force-free field that is ergodic on the surface of the domain. The proof involves three main ingredients: a gluing construction of piecewise smooth MHD equilibria, a Hamilton-Jacobi equation on the two-dimensional torus that can be understood as a nonlinear deformation of the cohomological equation (so the nondegeneracy assumption plays a major role in the corresponding analysis), and a new KAM theorem tailored for the study of divergence-free fields in three dimensions whose Poincaré map cannot be computed explicitly.
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