论文标题
在Woltjer的无力度最小化器和Moffatt的磁性放松上
On Woltjer's force free minimizers and Moffatt's magnetic relaxation
论文作者
论文摘要
在本说明中，我们表现出一种情况，即莫法特理想的磁性松弛问题的固定状态与Woltjer变分原理的相应无力量$ l^2 $能量最小化的情况不同。在莫法特（Moffatt）关于该主题的开创性工作中已经设想了此类例子，并涉及基本链接磁管的集合中支持的无分流矢量场。莫法特的例子的理由需要最小化序列的强烈收敛。目前的注释中证明的是，全球最小值（{\ em woltjer的最小化器}）与弱$ l^2 $闭合从拓扑范围内获得的矢量字段的最小值之间存在差距，这是通过能量偿还能量来偿还差异的矢量场的差异。在泰勒的猜想的背景下，我们的结果表明，如果初始场具有非平凡的拓扑结构，则无法在完美传导的磁富漏的粘性MHD松弛期间达到Woltjer的最小化器。该结果还适用于Moffatt的放松之外的任何其他放松过程，这些放松过程通过能量降低的差异性而发展出无分歧的领域，Vallis et.al和Nishiyama提出了此类过程。
In this note, we exhibit a situation where a stationary state of Moffatt's ideal magnetic relaxation problem is different than the corresponding force-free $L^2$ energy minimizer of Woltjer's variational principle. Such examples have been envisioned in Moffatt's seminal work on the subject and involve divergence free vector fields supported on collections of essentially linked magnetic tubes. Justification of Moffatt's examples requires the strong convergence of a minimizing sequence. What is proven in the current note is that there is a gap between the global minimum ({\em Woltjer's minimizer}) and the minimum over the weak $L^2$ closure of the class of vector fields obtained from a topologically non-trivial field by energy-decreasing diffeomorphisms. In the context of Taylor's conjecture, our result shows that the Woltjer's minimizer cannot be reached during the viscous MHD relaxation in the perfectly conducting magneto-fluid if the initial field has a nontrivial topology. The result also applies beyond Moffatt's relaxation to any other relaxation process which evolves a divergence free field by means of energy-decreasing diffeomorphisms, such processes were proposed by Vallis et.al and more recently by Nishiyama.
