Obstructions to Topological Relaxation for Generic Magnetic Fields

IF 2.6 1区 数学 Q1 MATHEMATICS, APPLIED
Alberto Enciso, Daniel Peralta-Salas
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引用次数: 0

Abstract

For any (analytic) axisymmetric toroidal domain \(\Omega \subset \mathbb {R}^3\) we prove that there is a locally generic set of divergence-free vector fields that are not topologically equivalent to any magnetohydrostatic (MHS) state in \(\Omega \). Each vector field in this set is Morse–Smale on the boundary, does not admit a nonconstant first integral, and exhibits fast growth of periodic orbits; in particular this set is residual in the Newhouse domain. The key dynamical idea behind this result is that a vector field with a dense set of nondegenerate periodic orbits cannot be topologically equivalent to a generic MHS state. On the analytic side, this geometric obstruction is implemented by means of a novel rigidity theorem for the relaxation of generic magnetic fields with a suitably complex orbit structure.

通用磁场拓扑弛豫的障碍
对于任意(解析的)轴对称环面域\(\Omega \subset \mathbb {R}^3\),我们证明了在\(\Omega \)中存在一个局部泛型的无散度矢量场集合,它们在拓扑上不等同于任何磁流体静力(MHS)状态。该集合中的每个向量场在边界上都是莫尔斯小的,不允许非常数第一积分,并且表现出周期轨道的快速增长;特别是这个集合在纽豪斯域中是残差的。该结果背后的关键动力学思想是,具有密集非简并周期轨道集的向量场在拓扑上不能等同于一般的MHS状态。在解析方面,利用具有适当复杂轨道结构的一般磁场弛豫的新刚性定理实现了这种几何障碍。
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来源期刊
CiteScore
5.10
自引率
8.00%
发文量
98
审稿时长
4-8 weeks
期刊介绍: The Archive for Rational Mechanics and Analysis nourishes the discipline of mechanics as a deductive, mathematical science in the classical tradition and promotes analysis, particularly in the context of application. Its purpose is to give rapid and full publication to research of exceptional moment, depth and permanence.
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