A Review of results on axially symmetric Navier-Stokes equations, with addendum by X. Pan and Q. S. Zhang

TL;DR

This paper surveys recent advances in axially symmetric Navier-Stokes equations, focusing on regularity, Liouville properties, decay, and new results in 3D solutions, including an addendum with recent generalizations.

Contribution

It provides a comprehensive review of recent results on ASNS and introduces new generalizations in the study of 3D solutions in the addendum.

Findings

Closing of the scaling gap for ASNS

Vanishing of homogeneous D solutions in 3D slabs

New generalizations of recent results in 3D Navier-Stokes

Abstract

In this paper, we give a brief survey of recent results on axially symmetric Navier-Stokes equations (ASNS) in the following categories: regularity criterion, Liouville property for ancient solutions, decay and vanishing of stationary solutions. Some discussions also touch on the full 3 dimensional equations. Two results, closing of the scaling gap for ASNS and vanishing of homogeneous D solutions in 3 dimensional slabs will be described in more detail. In the addendum, two new results in the 3rd category will also be presented, which are generalizations of recently published results by the author and coauthors.