论文标题
部分可观测时空混沌系统的无模型预测
Linearisability of divergence-free fields along invariant 2-tori
论文作者
论文摘要
我们发现条件下，可以限制无差的矢量字段$ b $对不变的环形表面$ s $的限制是可行的。主要结果与Arnold的结构定理相似，但所需的假设比换向$ [b，\ nabla \ times b] = 0 $要弱。放松对$ b $（也称为通量函数）的第一积分的需求，我们假设存在解决方案$ u：s \ to \ mathbb {r} $ to colomologice方程$ b | _s（u）= \ partial_n b $在$ s $ s $ s $ s $ b $ b $和$ b $和$ b $ \ nabla上\ b $ \ b $ b $ b $ b $ b $ b $ b $ b $右侧$ \ partial_n b $是可用于切换到$ s $的矢量字段的普通表面导数。在这种情况下，我们表明$ s $上的字段$ b $要么是零，要么用$ b | _s/\ | b \ |^2 | _s $可线化。我们将后者称为$ b $的半线性性性（具有比例$ \ | b \ |^2 | _s $）。非变化的特性依赖于BERS的伪分析功能理论，这些理论涉及由Witten共同体学变形引起的广义Laplace-Beltrami方程。通过使用De Rham的同时学，我们还指出了一个二元的积分条件，可以得出结论，$ b | _s $本身是可线的。 $ b | _s $的线性化性对于所谓的磁坐标至关重要，这对于磁性狭窄的等离子体理论至关重要。
We find conditions under which the restriction of a divergence-free vector field $B$ to an invariant toroidal surface $S$ is linearisable. The main results are similar in conclusion to Arnold's Structure Theorems but require weaker assumptions than the commutation $[B,\nabla\times B] = 0$. Relaxing the need for a first integral of $B$ (also known as a flux function), we assume the existence of a solution $u : S \to \mathbb{R}$ to the cohomological equation $B|_S(u) = \partial_n B$ on a toroidal surface $S$ mutually invariant to $B$ and $\nabla \times B$. The right hand side $\partial_n B$ is a normal surface derivative available to vector fields tangent to $S$. In this situation, we show that the field $B$ on $S$ is either identically zero or nowhere vanishing with $B|_S/\|B\|^2 |_S$ being linearisable. We are calling the latter the semi-linearisability of $B$ (with proportionality $\|B\|^2 |_S$). The non-vanishing property relies on Bers' results in pseudo-analytic function theory about a generalised Laplace-Beltrami equation arising from Witten cohomology deformation. With the use of de Rham cohomology, we also point out a Diophantine integral condition where one can conclude that $B|_S$ itself is linearisable. The linearisability of $B|_S$ is fundamental to the so-called magnetic coordinates, which are central to the theory of magnetically confined plasmas.
