An antisymmetric solution of the 3D incompressible Navier-Stokes equations

TL;DR

This paper investigates a specific incompressible Navier-Stokes flow in three dimensions, showing tornado-like behavior without actual blow-up, and discusses potential scenarios for true singularity formation.

Contribution

It provides a detailed analysis of a complex singular solution related to Li and Sinai's work, highlighting conditions that prevent real blow-up in 3D Navier-Stokes flows.

Findings

Flow exhibits tornado-like vorticity and velocity increase

No actual blow-up occurs due to approximate axial symmetry

Discussion on potential conditions for real blow-up

Abstract

We report results on the behavior of a particular incompressible Navier-Stokes (NS) flow in the whole space $\R^{3}$, related to the complex singular solutions introduced by Li and Sinai in \cite{LiSi08} that blow up at a finite time. The flow exhibits for some time a tornado-like behavior: a sharp increase of vorticity and of the maximal velocity, with concentration in an annular region around an axis. There is however no rotation around the axis and the approximate axial symmetry with no swirl excludes a real blow-up. We conclude with a discussion on the possible candidates for a real blow-up.