In this paper, I further continue an investigation on Beltrami Flows began in 2015 with A. Sorin and amply reprised and developed in 2022 with M. Trigiante. Instead of a compact $3$-torus $T^3=\mathbb{R}^3/Λ$ where $Λ$ is a crystallographic lattice, as done in previous work, here I considered flows confined in a cylinder with identified opposite bases. In this topology I considered axial symmetric flows and found a complete basis of axial symmetric harmonic $1$-forms that, for each energy level, decomposes into six components: two Beltrami, two anti-Beltrami and two closed forms. These objects, that are written in terms of trigonometric and Bessel functions, constitute a function basis for an $L^2$ space of axial symmetric flows. I have presented a general scheme for the search of axial symmetric solutions of Navier Stokes equation by reducing the latter to an hierachy of quadratic relations on the development coefficients of the flow in the above described functional basis. It is proposed that the coefficients can be determined by means of a Physics Informed like Neural Network optimization recursive algorithm. Indeed the present paper provides the theoretical foundations for such a algorithmic construction that is planned for a future publication.

翻译：本文延续了自2015年与A. Sorin合作启动、并于2022年与M. Trigiante系统拓展的Beltrami流研究。与先前工作中采用的紧致三维环面$T^3=\mathbb{R}^3/Λ$（其中$Λ$为晶体学格点）不同，本文考虑流体约束于两端面等同的圆柱形区域。在此拓扑结构下，我们研究了轴对称流动，并构建了一组完整的轴对称调和$1$-形式基。该基在每个能级下可分解为六个分量：两个Beltrami分量、两个anti-Beltrami分量及两个闭形式分量。这些由三角函数与贝塞尔函数表达的对象构成了轴对称流动$L^2$空间的函数基。通过将Navier-Stokes方程约化为关于流动在上述函数基中展开系数的二次关系递推体系，本文提出了寻找轴对称解的一般框架。建议采用类物理信息神经网络的优化递归算法确定展开系数。本文为后续拟发表的算法构建提供了理论基础。