论文标题
Beltrami流的典型野外线和Beltrami流的边界线线行为，简单地连接，紧凑，光滑的歧管与边界
Typical field lines of Beltrami flows and boundary field line behaviour of Beltrami flows on simply connected, compact, smooth manifolds with boundary
论文作者
论文摘要
我们表征了Beltrami流动在紧凑型，连接的歧管上的边界场线行为与第一个DE RHAM共同学组消失的歧管。也就是说，除了边界的密集子集最多无处，Beltrami田可能会消失，边界上的所有其他字段线都平稳地嵌入了$ 1 $ -Manifolds diffefemorphic到$ \ Mathbb {r} $，随着时间的推移，该零以$ \ pm \ pm \ infty $ \ mathbb {r} $。然后，我们删除了紧凑性并消失的De Rham的共同体学的假设，并证明，在给定的歧管上，几乎每个点，穿过该点的野外线是非恒定的，周期性的轨道或非周期性轨道，或者是一个非周期性的轨道，随着时间的推移，该轨道是任意接近起点的$ \ pm pm \ pm \ pm \ infty $。在证明过程中，我们将尤其表明，在歧管内部消失的Beltrami领域在Federer的意义上是$ 1 $的$ 1 $归档的集合，因此尤其是最多具有$ 1 $的Hausdorff尺寸。结果，我们得出的结论是，对于与非零特征值相对应的每个卷曲算子的特征菲尔德，总是存在一个鼻子域。
We characterise the boundary field line behaviour of Beltrami flows on compact, connected manifolds with vanishing first de Rham cohomology group. Namely we show that except for an at most nowhere dense subset of the boundary, on which the Beltrami field may vanish, all other field lines at the boundary are smoothly embedded $1$-manifolds diffeomorphic to $\mathbb{R}$, which approach the zero set as time goes to $\pm \infty$. We then drop the assumptions of compactness and vanishing de Rham cohomology and prove that for almost every point on the given manifold, the field line passing through the point is either a non-constant, periodic orbit or a non-periodic orbit which comes arbitrarily close to the starting point as time goes to $\pm \infty$. During the course of the proof we will in particular show that the set of points at which a Beltrami field vanishes in the interior of the manifold is countably $1$-rectifiable in the sense of Federer and hence in particular has a Hausdorff dimension of at most $1$. As a consequence we conclude that for every eigenfield of the curl operator, corresponding to a non-zero eigenvalue, there always exists exactly one nodal domain.
