On the Structure of Axisymmetric Helical Solutions to the Incompressible Navier–Stokes System

A class of exact solutions to the Navier–Stokes equations for an axisymmetric rotational incompressible flow is obtained. Invariant manifolds of flows that are axisymmetric about a given axis in three-dimensional coordinate space are found, and the structure of solutions is described. It is established that typical invariant regions of such flows are figures of rotation homeomorphic to the torus, which form a topological stratification structure, for example, in a ball, cylinder, and general complexes made up of such figures. The results extend to similar solutions of the system of MHD equations and Maxwell’s electrodynamic equations, which have analogous properties in \({{\mathbb{R}}_{3}}\) . Examples are given of axisymmetric vorticity vector fields and topological stratifications they generate on manifolds in \({{\mathbb{R}}_{3}}\) that are invariant under the dynamical systems specified by these fields.