On Liouville Type of Theorems to the 3-D Incompressible Axisymmetric Navier-Stokes Equations

TL;DR

This paper proves Liouville type theorems for 3D axisymmetric Navier-Stokes equations with swirl, under natural, scaling-invariant assumptions on the swirl component, aiding understanding of global regularity.

Contribution

It establishes new Liouville theorems for axisymmetric Navier-Stokes equations with swirl, under assumptions that are natural and scaling invariant, advancing the study of global regularity.

Findings

Liouville theorems are proved under specific assumptions on swirl.

Assumptions on $u_\theta$ are natural and scaling invariant.

Results facilitate further studies on global regularity.

Abstract

Liouville type of theorems play a key role in the blow-up approach to study the global regularity of the three-dimensional Navier-Stokes equations. In this paper, we will prove Liouville type of theorems to the 3-D axisymmetric Navier-Stokes equations with swirls under some suitable assumptions on swirl component velocity $u_\theta$ which are scaling invariant. It is known that $ru_\theta$ satisfies the maximum principle. The assumptions on $u_\theta$ will be natural and useful to make further studies on the global regularity to the three-dimensional incompressible axisymmetric Navier-Stokes equations.