论文标题
部分可观测时空混沌系统的无模型预测
Chaos from Symmetry: Navier Stokes equations, Beltrami fields and the Universal Classifying Crystallographic Group
论文作者
论文摘要
本文的核心是一种新颖的群体理论方法，由现任作者与亚历山大·索林（Alexander Sorin）合作于2015年发起，该方法允许在torii $ \ mathbb {r}^3/λ$ where $λ$的情况下，在torii $ \ mathbb {r}^3/λ$中进行更系统的分类和算法结构。基于通用分类组$ \ mathfrak {ug}_λ$的想法的新水合理论正在这里进行修订。我们构造了迄今为止缺少$ \ mathfrak {ug} _ {λ_{hex}} $，用于六边形晶格。掌握立方体和六角形实例，我们可以涵盖所有案件。 Beltrami流与接触结构之间的分类关系得到了启发。当前工作所开放的最有前途的研究方向是从一个事实流出的，即通用纳维尔 - 螺旋式解决方案的傅立叶系列扩展可以重新集成为无限的贡献$ \ m m i \ m i \ mathbf {w} _r $，每个贡献都与量子lati的量子和级别级别的量子级别的量子级r $相关的球形级别，并与baltim rattim and a the compaum and a baltim andim andim cantermi cantermi canterim andim cantermi canterim and cartimim r $相关。分解$ \ mathfrak {ug}_λ$的组的删除，它们在较高的层上重复。这种关键属性可以用规定的隐藏对称性作为NS方程的候选解决方案来构建通用傅立叶系列。 作为该研究计划的进一步结果，已构建了名为\ textbf {almafluidanspsystem}的Mathematica代码的完整而多功能的系统}，现在已构建，现在可以通过Wolfram社区的网站获得。从我们的构造中流的主要消息是，Beltrami流的对称性越多，最高的是混乱轨迹发作的概率。
The core of this paper is a novel group-theoretical approach, initiated in 2015 by one of the present authors in collaboration with Alexander Sorin, which allows for a more systematic classification and algorithmic construction of Beltrami flows on torii $\mathbb{R}^3/Λ$ where $Λ$ is a crystallographic lattice. The new hydro-theory, based on the idea of a Universal Classifying Group $\mathfrak{UG}_Λ$, is here revised. We construct the so far missing $\mathfrak{UG}_{Λ_{Hex}}$ for the hexagonal lattice. Mastering the cubic and hexagonal instances, we can cover all cases. The classification relation between Beltrami Flows and contact structures is enlightened. The most promising research direction opened by the present work streams from the fact that the Fourier series expansion of a generic Navier-Stokes solution can be regrouped into an infinite sum of contributions $\mathbf{W}_r$, each associated with a spherical layer of quantized radius $r$ in the momentum lattice and consisting of a superposition of a Beltrami and an anti-Beltrami field, with an analogous decomposition into irreps of the group $\mathfrak{UG}_Λ$ that are variously repeated on higher layers. This crucial property enables the construction of generic Fourier series with prescribed hidden symmetries as candidate solutions of the NS equations. As a further result of this research programme a complete and versatile system of MATHEMATICA Codes named \textbf{AlmafluidaNSPsystem} has been constructed and is now available through the site of the Wolfram Community. The main message streaming from our constructions is that the more symmetric the Beltrami Flow the highest is the probability of the onset of chaotic trajectories.
