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Volume 38, Issue 3
A Review of Results on Axially Symmetric Navier-Stokes Equations, with Addendum by X. Pan and Q. Zhang

Qi Zhang & Xinghong Pan

Anal. Theory Appl., 38 (2022), pp. 243-296.

Published online: 2022-07

[An open-access article; the PDF is free to any online user.]

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  • 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.

  • Keywords

Regularity, Liouville theorem, ancient solutions, D-solutions, axially symmetric Navier-Stokes equations.

  • AMS Subject Headings

35Q30, 76N10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-38-243, author = {Qi and Zhang and and 24106 and and Qi Zhang and Xinghong and Pan and and 24107 and and Xinghong Pan}, title = {A Review of Results on Axially Symmetric Navier-Stokes Equations, with Addendum by X. Pan and Q. Zhang}, journal = {Analysis in Theory and Applications}, year = {2022}, volume = {38}, number = {3}, pages = {243--296}, 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.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-2020-0050}, url = {http://global-sci.org/intro/article_detail/ata/20802.html} }
TY - JOUR T1 - A Review of Results on Axially Symmetric Navier-Stokes Equations, with Addendum by X. Pan and Q. Zhang AU - Zhang , Qi AU - Pan , Xinghong JO - Analysis in Theory and Applications VL - 3 SP - 243 EP - 296 PY - 2022 DA - 2022/07 SN - 38 DO - http://doi.org/10.4208/ata.OA-2020-0050 UR - https://global-sci.org/intro/article_detail/ata/20802.html KW - Regularity, Liouville theorem, ancient solutions, D-solutions, axially symmetric Navier-Stokes equations. AB -

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.

Qi Zhang & Xinghong Pan. (2022). A Review of Results on Axially Symmetric Navier-Stokes Equations, with Addendum by X. Pan and Q. Zhang. Analysis in Theory and Applications. 38 (3). 243-296. doi:10.4208/ata.OA-2020-0050
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