关于Woltjer的无力极小元和Moffatt的磁弛豫
On Woltjer's force free minimizers and Moffatt's magnetic relaxation
摘要：
在这篇文章中，我们展示了一种情况下，Moffatt的理想磁弛豫问题的一个定态是不同于相应的无力$L^2$能量最小的Woltjer的变分原理。这样的例子已经设想在莫法特的开创性工作的主题，并涉及发散自由向量场支持的集合基本上链接的磁管。证明莫法特的例子需要最小化序列的强收敛性。本文证明了由拓扑非平凡域通过能量减同态得到的向量场类的全局极小值（{ \em Woltjer极小值}）与弱$L^2$闭包上的极小值之间存在一个间隙.在泰勒猜想的背景下，我们的结果表明，Woltjer的最小值不能达到粘性MHD松弛过程中的理想导电磁流体，如果初始场具有非平凡的拓扑结构。结果也适用于超越莫法特的松弛到任何其他松弛过程，演变的发散自由场的能量减少的同态，这样的过程提出了瓦利斯et.al和最近由西山。
英文摘要：
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.
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