Structure of Singularities of 3D Axi-symmetric Navier-Stokes Equations

TL;DR

This paper analyzes the local structure of potential singularities in axially symmetric Navier-Stokes solutions, showing they resemble constant flows under certain scaling conditions near points of high flow speed.

Contribution

It characterizes the local behavior near singularities in 3D axially symmetric Navier-Stokes flows, demonstrating convergence to constant solutions after appropriate scaling.

Findings

Solutions near potential singularities resemble constant flows after scaling.

High flow speed points are locally close to nonzero constant vectors.

Results apply under conditions relating flow speed and distance from the axis.

Abstract

Let $v$ be a solution of the axially symmetric Navier-Stokes equation. We determine the structure of certain (possible) maximal singularity of $v$ in the following sense. Let $(x_0, t_0)$ be a point where the flow speed $Q_0 = |v(x_0, t_0)|$ is comparable with the maximum flow speed at and before time $t_0$. We show after a space-time scaling with the factor $Q_0$ and the center $(x_0, t_0)$, the solution is arbitrarily close in $C^{2, 1, \alpha}_{{\rm local}}$ norm to a nonzero constant vector in a fixed parabolic cube, provided that $r_0 Q_0$ is sufficiently large. Here $r_0$ is the distance from $x_0$ to the $z$ axis. Similar results are also shown to be valid if $|r_0v(x_0, t_0)|$ is comparable with the maximum of $|rv(x, t)|$ at and before time $t_0$.