Liouville theorem of axially symmetric Navier-Stokes equations with growing velocity at infinity

TL;DR

This paper extends Liouville theorems for axially symmetric Navier-Stokes solutions, allowing solutions to grow sublinearly at infinity, and demonstrates the optimality of this growth restriction with counterexamples.

Contribution

It improves previous Liouville theorems by permitting solutions with sublinear growth and establishes the sharpness of this growth condition.

Findings

Liouville theorem holds for solutions with growth less than linear

Counterexamples show failure of the theorem at linear growth

Optimal growth rate for the theorem is identified as sublinear

Abstract

In the paper \cite{KNSS:1}, the authors make the following conjecture: {\it any bounded ancient mild solution of the 3D axially symmetric Navier-Stokes equations is constant.} And it is proved in the case that the solution is swirl free. Our purpose of this paper is to improve their result by allowing that the solution can grow with a power smaller than 1 with respect to the distance to the origin. Also, we will show that such a power is optimal to prove the Liouville type theorem since we can find counterexamples for the Navier-Stokes equations such that the Liouville theorem fails if the solution can grow linearly.