论文标题
高维
Steady Euler flows and Beltrami fields in high dimensions
论文作者
论文摘要
使用开放书籍，我们证明了在任何奇数歧管的每个非变化矢量场的同型类别中，对于某些指标的Euler方程来说，存在非逐渐稳定的解决方案。作为推论，任何此类领域都可以在高尺寸的球体上的触点Reeb场的不变子序列中实现。构造的溶液是可去毒的，因此可以修改Beltrami型，以获得混乱的流体。我们在奇数方面表征了Beltrami字段，并表明始终存在具有数量的Beltrami字段，这些字段既不是地理位置的，也不是Euler流动的任何度量。这与三维情况形成鲜明对比，在三维情况下，每个体积保存的Beltrami场都是某些度量的稳定的欧拉流动。最后，我们在每个奇数大于三的奇数中都没有周期性的轨道构建一个不散布的Beltrami场（不一定是容量的）。
Using open books, we prove the existence of a non-vanishing steady solution to the Euler equations for some metric in every homotopy class of non-vanishing vector fields of any odd dimensional manifold. As a corollary, any such field can be realized in an invariant submanifold of a contact Reeb field on a sphere of high dimension. The constructed solutions are geodesible and hence of Beltrami type, and can be modified to obtain chaotic fluids. We characterize Beltrami fields in odd dimensions and show that there always exist volume-preserving Beltrami fields which are neither geodesible nor Euler flows for any metric. This contrasts with the three dimensional case, where every volume-preserving Beltrami field is a steady Euler flow for some metric. Finally, we construct a non-vanishing Beltrami field (which is not necessarily volume-preserving) without periodic orbits in every manifold of odd dimension greater than three.
