Exact Solutions on Twisted Rings for the 3D Navier-Stokes Equations

TL;DR

This paper constructs explicit, regular solutions to the 3D Navier-Stokes equations on a twisted vortex ring, providing insights into solution behavior and narrowing the search for potential blow-up counterexamples.

Contribution

It presents the first explicit solutions on twisted rings for the 3D Navier-Stokes equations, enhancing understanding of solution regularity and vortex dynamics.

Findings

Solutions remain regular at the vortex core

Flow exhibits rotation and swirling with spiraling flux

Results suggest constraints on finite-time blow-up scenarios

Abstract

The problem of describing the behavior of the solutions to the Navier-Stokes equations in three space dimensions has always been borderline. From one side, due to the viscosity term, smooth data seem to produce solutions with an everlasting regular behavior. On the other hand, the lack of a convincing theoretical analysis suggests the existence of possible counterexamples. In particular, one cannot exclude the blowing up of solutions in finite time even in presence of smooth data. Here we give examples of explicit solutions of the non-homogeneous equations. These are defined on a Hill's type vortex where the flow is rotating and swirling at the same time, inducing the flux to spiraling at a central node. Despite the appearance, the solution still remains very regular at the agglomeration point. The analysis may lead to a better understanding of the subtle problem of characterizing the solution space of the 3D Navier-Stokes equations. For instance, this result makes more narrow the path to the search of counterexamples built on the stretching and twisting of vortex tubes.