Chaos from Symmetry: Navier Stokes Equations, Beltrami Fields and the Universal Classifying Crystallographic Group

Abstract

The core of this paper is the group-theoretical approach, initiated in 2015 by one of the present authors in collaboration with Alexander Sorin that brings into the classical field of mathematical fluid-mechanics a brand new vision, allowing for a more systematic classification and algorithmic construction of Beltrami flows on torii R3/Λ where Λ is a crystallographic lattice. Here this new hydro-theory based on the focal idea of a Universal Classifying Group UGΛ is revised, reorganized, improved and extended. In particular we construct the so far missing UGΛHex for the hexagonal lattice and we advocate that, mastering the cubic and hexagonal instances of this group, we can cover all cases. The relation between the classification of Beltrami Flows with that of contact structures is enlightened. The recent developments about the framework of b-manifolds are considered and it is shown that the choice of the allowed critical surfaces for the b-deformation of a Beltrami field seems to be strongly related to the group-theoretical structure of the latter. This opens new directions of investigations about a group theoretical classification of critical surfaces. Apart from that 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 Wr, each associated with a spherical layer of quantized radius r in the momentum lattice. Each Wr is the superposition of a Beltrami field Wr+ plus an anti-Beltrami field Wr−. These latter have a priori exactly the same decomposition into irreps of the group UGΛ 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. Alternatively the Fourier series representation of known solutions can be analyzed from the point of view of such symmetries. As a further result of this research program a complete and versatile system of MATHEMATICA Codes named AlmafluidaNSPsystem has been constructed and is now available through the site of the Wolfram Community. The exact solutions presented in this paper have to be considered as an illustration of the new conceptions and ideas that have emerged and of what can be further done utilizing the computer codes as an instrument. The main message streaming from our constructions is that the more symmetric is the Beltrami Flow the highest is the probability of the on-set of chaotic trajectories.