Axial symmetric Navier Stokes equations and the Beltrami/antiBeltrami spectrum in view of Physics Informed Neural Networks

Abstract In this paper, I further continue an investigation on Beltrami Flows began in 2015 with A. Sorin and amply revised and developed in 2022 with M. Trigiante. Instead of a compact 3-torus T 3 = R 3 / Λ ${T}^{3}={\mathbb{R}}^{3}/{\Lambda}$ 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 antiBeltrami 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 a 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 an algorithmic construction that is planned for a future publication.