论文标题
嵌套不变的托里（Tori）叶出现矢量场及其卷发：朝向MHD平衡和稳定的欧拉在没有连续的欧几里得异分析的环形域中流动
Nested invariant tori foliating a vector field and its curl: toward MHD equilibria and steady Euler flows in toroidal domains without continuous Euclidean isometries
论文作者
论文摘要
本文研究了找到三维螺线管载体场的问题，以使矢量场及其卷曲都与给定的环形表面家族相切。我们表明，可以将这个问题转化为确定具有二维线性椭圆二阶二阶偏微分方程的周期性溶液的问题，并证明平滑溶液的存在。在连续的欧几里得异构体下不变的环形表面散发出的平滑溶液的实例也被明确构建，并且被鉴定为各向异性磁流失动力学的平衡。此处检查的问题代表了基本数学问题的较弱版本，该问题是在磁性水力动力学和流体力学的背景下，涉及存在常规平衡磁场和没有连续欧几里得异构体的有限域中的稳定欧拉流。这种构型的存在代表了核融合反应器中限制磁场设计的关键理论问题，称为恒星。
This paper studies the problem of finding a three-dimensional solenoidal vector field such that both the vector field and its curl are tangential to a given family of toroidal surfaces. We show that this question can be translated into the problem of determining a periodic solution with periodic derivatives of a two-dimensional linear elliptic second-order partial differential equation on each toroidal surface, and prove the existence of smooth solutions. Examples of smooth solutions foliated by toroidal surfaces that are not invariant under continuous Euclidean isometries are also constructed explicitly, and they are identified as equilibria of anisotropic magnetohydrodynamics. The problem examined here represents a weaker version of a fundamental mathematical problem that arises in the context of magnetohydrodynamics and fluid mechanics concerning the existence of regular equilibrium magnetic fields and steady Euler flows in bounded domains without continuous Euclidean isometries. The existence of such configurations represents a key theoretical issue for the design of the confining magnetic field in nuclear fusion reactors known as stellarators.
