Singular stationary Navier-Stokes flows: examples and stability
Abstract
Landau solutions, when oriented along the vertical axis, represent a one parameter family of exact, self-similar, axisymmetric, swirl-free solutions to the 3D stationary Navier-Stokes equations forced by an upward facing point-source of momentum at the origin. They have an isolated singularity at the origin. Other singular steady state solutions can be derived from a similar framework. Some of these variants have been proposed as models for physical scenarios. For example, a solution first found by Squire has been proposed as a model of a fluid entrained to a radially discharging surface layer of oil. Another, Serrin's swirling vortex, exhibits qualitative features shared with some tornadoes, like a two-cell structure consisting of a central downdraft and peripheral updraft as well as swirl. The first objective of this paper is to provide a detailed analysis of these and other examples, especially when boundaries are present. In this direction we find several new, physically motivated classes of solutions and identify new connections between the physics literature and the mathematics literature. Many of these examples are formulated on the half-space, but much of the mathematical literature on singular steady-state solutions is for ℝ3. The second objective of this paper is to establish asymptotic stability for many of these solutions by formulating the problem in domains with boundaries. Our most general result in this direction uses a new approach to asymptotic stability that is based on eventual regularity. In a different direction, we prove that a class of singular steady-state solutions motivated by Serrin's swirling vortex are stable under certain axisymmetric perturbations. This requires a novel observation because these ``Type III'' solutions are too singular to be directly amenable to existing approaches.
Related
- Liouville theorems for the stationary Navier-Stokes equations under partial component conditions
- uniqueness and asymptotic behavior of the stationary Navier-Stokes equations in a slab
- Liouville Theorems for Stationary Navier-Stokes Equations via the Radial Velocity Component
- On the effects of rotation on stationary Navier--Stokes equations in the half spaces in scale-critical spaces
- Self-similar stationary Navier--Stokes flows in a half-space with arbitrarily large swirl forcing
BibTeX
@misc{bradshaw2026singular,
title = {{Singular stationary Navier-Stokes flows: examples and stability}},
author = {Zachary Bradshaw and Dakota Palmer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.22291},
eprint = {2606.22291},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.22291},
}
RIS
TY - UNPB
TI - Singular stationary Navier-Stokes flows: examples and stability
AU - Zachary Bradshaw
AU - Dakota Palmer
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2606.22291
UR - https://arxiv.org/abs/2606.22291
AB - Landau solutions, when oriented along the vertical axis, represent a one parameter family of exact, self-similar, axisymmetric, swirl-free solutions to the 3D stationary Navier-Stokes equations forced by an upward facing point-source of momentum at the origin. They have an isolated singularity at the origin. Other singular steady state solutions can be derived from a similar framework. Some of these variants have been proposed as models for physical scenarios. For example, a solution first found by Squire has been proposed as a model of a fluid entrained to a radially discharging surface layer of oil. Another, Serrin's swirling vortex, exhibits qualitative features shared with some tornadoes, like a two-cell structure consisting of a central downdraft and peripheral updraft as well as swirl. The first objective of this paper is to provide a detailed analysis of these and other examples, especially when boundaries are present. In this direction we find several new, physically motivated classes of solutions and identify new connections between the physics literature and the mathematics literature. Many of these examples are formulated on the half-space, but much of the mathematical literature on singular steady-state solutions is for ℝ^3. The second objective of this paper is to establish asymptotic stability for many of these solutions by formulating the problem in domains with boundaries. Our most general result in this direction uses a new approach to asymptotic stability that is based on eventual regularity. In a different direction, we prove that a class of singular steady-state solutions motivated by Serrin's swirling vortex are stable under certain axisymmetric perturbations. This requires a novel observation because these ``Type III'' solutions are too singular to be directly amenable to existing approaches.
ER -