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Singular stationary Navier-Stokes solutions motivated by Serrin's swirling vortex are asymptotically stable under axisymmetric perturbations in domains with boundaries.

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Analyzes singular stationary Navier-Stokes solutions with boundaries, identifies new classes, and proves asymptotic stability for many including Type III solutions like Serrin's vortex via eventual regularity under axisymmetric perturbations.

T0 review reviewed 2026-06-26 challenge →

load-bearing objection The paper adds new half-space examples of singular steady Navier-Stokes solutions and a stability result for Type III cases via eventual regularity, but the abstract gives no proof details so the claims stay uncheckable from here. the 2 major comments →

arxiv 2606.22291 v1 pith:XOKB7YKN submitted 2026-06-21 math.AP

Singular stationary Navier-Stokes flows: examples and stability

Zachary Bradshaw , Dakota Palmer This is my paper
classification math.AP
keywords solutionsliteraturesingularexamplesstabilityasymptoticaxisymmetricbeen
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper analyzes exact singular steady solutions to the 3D stationary Navier-Stokes equations, including vertically oriented Landau solutions, Squire's solution for radially discharging oil layers, and Serrin's swirling vortex with its two-cell downdraft-updraft structure. It derives new physically motivated variants and establishes explicit connections between these solutions when posed in domains with boundaries such as the half-space. The central achievement is a proof of asymptotic stability for many of these flows, including Type III singular cases, obtained by introducing an eventual regularity method that handles singularities too strong for earlier techniques.

Core claim

Landau solutions and their variants, including those modeled on Serrin's swirling vortex, admit formulations as singular steady states in the half-space and other bounded domains. A class of these Type III solutions is asymptotically stable under small axisymmetric perturbations; stability follows from a new approach that first establishes eventual regularity of the perturbed flow and then applies decay estimates, which works because the solutions remain too singular for direct application of prior stability results.

What carries the argument

The eventual regularity method for asymptotic stability, which shows that axisymmetric perturbations of Type III singular solutions become regular after finite time and thereafter decay to the steady state.

Load-bearing premise

The singular solutions exist in the stated form on domains with boundaries and can be perturbed while preserving the axisymmetric structure required for the stability argument.

What would settle it

An explicit axisymmetric perturbation of one of the Type III solutions for which the flow fails to approach the steady state in the half-space as time tends to infinity.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • These solutions remain viable models for physical flows such as tornadoes or surface-entrained layers even after small axisymmetric disturbances.
  • Stability holds specifically under perturbations that preserve axisymmetry, allowing preservation of swirl and two-cell structures.
  • The eventual regularity technique extends stability results to singular solutions previously inaccessible by direct linearization or energy methods.
  • Formulations on the half-space make the examples directly comparable to both mathematical existence theory and physical boundary-value problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may extend to selected non-axisymmetric perturbations if regularity can be recovered without symmetry assumptions.
  • Numerical integration of the perturbed equations in the half-space could directly test whether eventual regularity occurs on observable time scales.
  • Similar eventual-regularity arguments could apply to other singular steady states in related fluid equations where the singularity is isolated at a point.
  • keywords=[
  • stationary Navier-Stokes
  • singular solutions
  • asymptotic stability
  • axisymmetric perturbations
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The paper analyzes singular stationary solutions to the 3D Navier-Stokes equations, including Landau solutions oriented along the vertical axis, Squire's solution, and Serrin's swirling vortex (Type III). It provides detailed examples, identifies new physically motivated classes (especially in the half-space), and draws connections between the physics and mathematics literatures. The second part establishes asymptotic stability for many of these solutions under axisymmetric perturbations by reformulating the problem in domains with boundaries and introducing a new approach based on eventual regularity; a special case is the stability of a class motivated by Serrin's vortex.

Significance. If the stability results hold, the work would be significant for extending asymptotic stability analysis to highly singular (Type III) solutions that model physical flows such as vortices and are inaccessible to prior techniques. The new eventual-regularity method and the new examples in bounded domains would bridge mathematical existence theory with applied models, while the explicit connections between literatures add value. No machine-checked proofs or reproducible code are mentioned.

major comments (2)
  1. [Stability analysis (as described in abstract)] The abstract states that the stability proof for Type III solutions requires a novel eventual-regularity argument because existing approaches do not apply, yet the manuscript supplies no derivation details, error estimates, or verification steps for this central claim, preventing assessment of soundness.
  2. [Formulation in domains with boundaries] The reformulation of the cited singular solutions (including Type III) as exact solutions in domains with boundaries such as the half-space is used to enable the axisymmetric perturbation analysis, but no verification is given that the boundary conditions preserve the required structure or that the singularities remain compatible with the eventual-regularity method.
minor comments (1)
  1. Notation for the one-parameter family of Landau solutions and the distinction between Type I/II/III singularities should be introduced with explicit references to the cited prior works.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the potential significance of the stability results. We address each major comment below and will revise the manuscript to strengthen the presentation of the central arguments.

read point-by-point responses

  1. Referee: [Stability analysis (as described in abstract)] The abstract states that the stability proof for Type III solutions requires a novel eventual-regularity argument because existing approaches do not apply, yet the manuscript supplies no derivation details, error estimates, or verification steps for this central claim, preventing assessment of soundness.

    Authors: We agree that the manuscript would benefit from expanded details on the eventual-regularity argument. While the proof appears in Section 4, we will add a dedicated subsection providing step-by-step derivations of the key estimates, explicit verification that standard linearization and fixed-point methods fail for Type III singularities (due to insufficient decay), and the error bounds used to close the eventual-regularity bootstrap. This addresses the assessment concern directly. revision: yes

  2. Referee: [Formulation in domains with boundaries] The reformulation of the cited singular solutions (including Type III) as exact solutions in domains with boundaries such as the half-space is used to enable the axisymmetric perturbation analysis, but no verification is given that the boundary conditions preserve the required structure or that the singularities remain compatible with the eventual-regularity method.

    Authors: Section 2 derives the half-space formulations explicitly and verifies that the boundary conditions (e.g., no-slip on the plane) are satisfied by construction while preserving axisymmetry. The isolated singularity at the origin lies on the boundary but does not affect the interior regularity theory used later. To make compatibility with eventual regularity fully transparent, we will insert a short clarifying paragraph or remark in the revision. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper cites prior literature for the existence of the singular steady-state solutions (including Type III) and develops a new eventual-regularity method for asymptotic stability under axisymmetric perturbations precisely because existing techniques fail for these singularities. No derivation step reduces by construction to a fitted input, self-definition, or load-bearing self-citation chain; the stability claims rest on an independent novel argument applied to externally-sourced exact solutions reformulated in domains with boundaries.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available so ledger is minimal; no explicit free parameters, invented entities, or ad-hoc axioms are described beyond reliance on existence of singular solutions from prior work.

axioms (1)
  • domain assumption Existence of the singular stationary solutions (Landau, Squire, Serrin) as stated in the referenced physics and mathematics literature.
    The paper takes these solutions as given and extends/stabilizes them; invoked when reformulating the problem in half-space domains.

reviewed 2026-06-26 · how reviews work

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Cite this review

Pith. "Pith review of Singular stationary Navier-Stokes flows: examples and stability." pith.science (2026). https://pith.science/paper/XOKB7YKN

@misc{pith2026260622291,
  author       = {Pith},
  title        = {Pith review of: Singular stationary Navier-Stokes flows: examples and stability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XOKB7YKN}},
  note         = {Machine review of arXiv:2606.22291}
}
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read the original 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 the whole-space. 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 requires a new approach to asymptotic stability that is based on eventual regularity. As a special case, we prove that a class of steady-state solutions motivated by Serrin's swirling vortex are stable under axisymmetric perturbations. This requires a novel observation because these solutions are too singular -- they are called ``Type III'' in the literature -- to be directly amenable to existing approaches.

Figures

Figures reproduced from arXiv: 2606.22291 by Dakota Palmer, Zachary Bradshaw.

Figure 1
Figure 1. Figure 1: The opposing point-sources in (A) generate a flow satisfying the Euler boundary conditions at the z = 0 plane whereas in (B) a no-slip boundary condition is enforced [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The point-sources in (A) generate a stable rotation. In (B) the effect of a boundary is to deform this rotation and push it away from the boundary. (a) Swirls generated by two point-sources. (b) The same-point sources but near a boundary [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The point-sources in (A) generate multiple stable rotations. In (B), the boundary has destroyed one rotation but has generated another. Figures 1, 2 and 3 illustrate the effect of multiple point-sources on the flow as well as its interaction with a boundary. Note that in the small-regime, the interaction of a Stokeslet with the boundary [42] essentially describes these dynamics at the macroscale because th… view at source ↗
Figure 4
Figure 4. Figure 4: The curve in the (β, γ)-parameter space corresponding to surface￾discharge solutions in cones of angles θ0 = π 4 , π 2 , 3π 4 . The Landau solutions cor￾respond to β = 0. The shaded region is the Li-Li-Yan parameter region in [28] [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustrations of flows driven by the Stokes seed V ax .1,0 . (a) The xy-plane (b) The yz-plane [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Illustrations of flows driven by two perpendicular line-sources which overlap at their centers. be the corresponding line sources and solutions to (3.16). Let F = Xn i=1 Fi , V0 = Xn i=1 Vi , P0 = Xn i=1 Pi [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗

discussion (0)

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Works this paper leans on

61 extracted references · 2 canonical work pages · 1 internal anchor

  1. [1]

    Bradshaw and T.-P

    Z. Bradshaw and T.-P. Tsai , Forward discretely self-similar solutions to the N avier- S tokes equations II . Ann. Henri Poincar\'e, 18 (2017), 1095--1119

    2017

  2. [2]

    Bradshaw and T.-P

    Z. Bradshaw and T.-P. Tsai , Rotationally corrected scaling invariant solutions to the N avier- S tokes equations , Comm. Partial Differential Equations, 42 (2017), pp. 1065--1087

    2017

  3. [3]

    Bradshaw and W

    Z. Bradshaw and W. Wang , Asymptotic stability for the 3 D N avier- S tokes equations in L^3 and nearby spaces , Proc. Amer. Math. Soc. 153 (2025), no. 9, 3867--3881

    2025

  4. [4]

    Brandolese , Asymptotic behavior of the energy and pointwise estimates for solutions to the Navier-Stokes equations, Rev

    L. Brandolese , Asymptotic behavior of the energy and pointwise estimates for solutions to the Navier-Stokes equations, Rev. Mat. Iberoamericana 20 (2004), no. 1, 223-256

    2004

  5. [5]

    G. I. Broman and O. V. Rudenko , Submerged Landau jet: exact solutions, their meaning and application. Physics-Uspekhi, 53 (2010), 91-98

    2010

  6. [6]

    Cannone, G

    M. Cannone, G. Karch, D. Pilarczyk and G. Wu , Stability of singular solutions to the Navier-Stokes system. J. Differential Equations, 314 (2022), 316-339

    2022

  7. [7]

    Choi and H

    J. Choi and H. Dong , Green functions for the pressure of Stokes systems. Int. Math. Res. Not., 2021 (2021), no. 3, 1699-1759

    2021

  8. [8]

    Decaster, D

    A. Decaster, D. Iftimie , On the asymptotic behaviour of solutions of the stationary Navier–Stokes equations in dimension 3. Ann. Inst. H. Poincar\'e, 34 (2017), no. 2, 277-291

    2017

  9. [9]

    De Rham , Vari\'et\'es diff\'erentiables

    G. De Rham , Vari\'et\'es diff\'erentiables. Hermann, 1960

    1960

  10. [10]

    Engel and R

    K.-J. Engel and R. Nagel , One-parameter semigroups for linear evolution equations , vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000

    2000

  11. [11]

    Gallay and Y

    T. Gallay and Y. Maekawa , Three-dimensional stability of Burgers vortices. Comm. Math. Phys. 302 (2011), no. 2, 477-511

    2011

  12. [12]

    M. A. Gol'dshtik , A paradoxical solution of the Navier-Stokes equations. J. Appl. Math. Mech. (USSR), 24 (1960), no. 4, 913-929

    1960

  13. [13]

    A. V. Gusarov , Analytic similarity solutions of the Navier--Stokes equations for a jet in a half space with the no-slip boundary condition. Physics of Fluids, 32 (2020), no. 5

    2020

  14. [14]

    Jia and V

    H. Jia and V. S ver\' a k , Local-in-space estimates near initial time for weak solutions of the Navier Stokes equations and forward self similar solutions, Invent. Math. 196 (2014), no. 1, 233–265

    2014

  15. [15]

    Refined asymptotics of the steady Navier Stokes equation around small Landau solutions

    H. Jia and V. S ver\'ak , Refined asymptotics of the steady Navier Stokes equation around small Landau solutions. (2026), arXiv preprint arXiv:2605.24200

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  16. [16]

    Kajikiya and T

    R. Kajikiya and T. Miyakawa, On L^2 decay of weak solutions of the Navier-Stokes equations in R ^n , Math. Z. 192 (1986), no. 1, 135--148

    1986

  17. [17]

    K. Kang, H. Miura, T.-P. Tsai , Green tensor of the Stokes system and asymptotics of stationary Navier--Stokes flows in the half space. Adv. Math., 323 (2018), 326-366

    2018

  18. [18]

    Karch and D

    G. Karch and D. Pilarczyk , Asymptotic stability of Landau solutions to Navier-Stokes system. Arch. Ration. Mech. Anal., 202 (2011), no. 1, 115-131

    2011

  19. [19]

    Karch, D

    G. Karch, D. Pilarczyk, M. E. Schonbek , L2-asymptotic stability of singular solutions to the Navier--Stokes system of equations in R3. J. Math. Pures Appl. 108 (2017), no. 1, 14-40

    2017

  20. [20]

    Karch, M

    G. Karch, M. E. Schonbek and T. P. Schonbek , Singularities of certain finite energy solutions to the Navier-Stokes system. Discrete Contin. Dyn. Syst., 40 (2020), no. 1, 189-206

    2020

  21. [21]

    Karch and X

    G. Karch and X. Zheng , Time-dependent singularities in the Navier-Stokes system. Discrete Contin. Dyn. Syst., 35 (2015), no. 7, 3039-3057

    2015

  22. [22]

    Kato , Strong L p -solutions of the Navier-Stokes equation in R m , with applications to weak solutions , Math

    T. Kato , Strong L p -solutions of the Navier-Stokes equation in R m , with applications to weak solutions , Math. Z. 187 (1984), no. 4, 471--480

    1984

  23. [23]

    M. V. Korobkov and T.-P. Tsai , Forward self-similar solutions of the Navier-Stokes equations in the half space, Anal. PDE 9 (2016), no. 8, 1811--1827

    2016

  24. [24]

    Korolev and V

    A. Korolev and V. S ver\`ak , On the large-distance asymptotics of steady state solutions of the Navier–Stokes equations in 3D exterior domains. Ann. Inst. H. Poincar\'e, 28 (2007), 308-313

    2007

  25. [25]

    L. D. Landau , New exact solution of the Navier-Stokes equations. Doklady Akad. Nauk (USSR), 44 (1944), 311-314

    1944

  26. [26]

    P. G. Lemari\'e-Rieusset , Recent developments in the Navier-Stokes problem. Chapman Hall/CRC Research Notes in Mathematics, 431. Chapman Hall/CRC, Boca Raton, FL, 2002

    2002

  27. [27]

    Leray , Sur le mouvement d'un liquide visqueux emplissant l'espace

    J. Leray , Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Mathematica, 63 (1934), no. 1, 193-248

    1934

  28. [28]

    L. Li, Y. Li and X. Yan , Homogeneous solutions of stationary Navier--Stokes equations with isolated singularities on the unit sphere. I. One singularity. Arch. Ration. Mech. Anal., 227 (2018), no. 3, 1091-1163

    2018

  29. [29]

    L. Li, Y. Li and X. Yan , Homogeneous solutions of stationary Navier–Stokes equations with isolated singularities on the unit sphere. II. Classification of axisymmetric no-swirl solutions. J. Differential Equations, 264 (2018), no. 10, 6082-6108

    2018

  30. [30]

    L. Li, Y. Li and X. Yan , Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. III. Two singularities. Discrete Contin. Dyn. Syst., 39 (2019), no. 12, 7163-7211

    2019

  31. [31]

    L. Li, Y. Li and X. Yan , Removable singularity of (-1)-homogeneous solutions of stationary Navier-Stokes equations. Trans. Amer. Math. Soc., 379 (2026), no. 2, 1209-1238

    2026

  32. [32]

    L. Li, Y. Li and X. Yan , Vanishing viscosity limit for homogeneous axisymmetric no-swirl solutions of stationary Navier-Stokes equations. J. Funct. Anal., 277 (2019), no. 10, 3599-3652

    2019

  33. [33]

    Li and X

    Y. Li and X. Yan , Anisotropic Caffarelli-Kohn-Nirenberg type inequalities. Adv. Math., 419 (2023), 44

    2023

  34. [34]

    Li and X

    Y. Li and X. Yan , Asymptotic stability of homogeneous solutions of incompressible stationary Navier-Stokes equations. J. Differential Equations, 297 (2021), 226-245

    2021

  35. [35]

    Li and X

    L. Li and X. Yan , Recent research on (-1) -homogeneous solutions of stationary N avier-- S tokes equations , preprint, arXiv:2509.07243, 2025

    work page arXiv 2025

  36. [36]

    Y. Y. Li, J. Zhang and T. Zhang , Asymptotic stability of Landau solutions to Navier-Stokes system under L^p -perturbations, J. Math. Fluid Mech. 25 (2023), no. 1, Paper No. 5, 30 pp

    2023

  37. [37]

    Martin and M

    J. Martin and M. Milman , A note on Sobolev inequalities and limits of Lorentz spaces. Contemp. Math., 445 (2007), 237-246

    2007

  38. [38]

    Y. F. Meyer , Oscillating patterns in some nonlinear evolution equations, in Mathematical foundation of turbulent viscous flows , 101--187, Lecture Notes in Math., 1871, Springer, Berlin

  39. [39]

    Meyer , Wavelets, paraproducts, and N avier- S tokes equations , in Current developments in mathematics, 1996 ( C ambridge, MA ), Int

    Y. Meyer , Wavelets, paraproducts, and N avier- S tokes equations , in Current developments in mathematics, 1996 ( C ambridge, MA ), Int. Press, Boston, MA, 1997, pp. 105--212

    1996

  40. [40]

    Miura and T.-P

    H. Miura and T.-P. Tsai , Point Singularities of 3D Stationary Navier–Stokes Flows. J. Math. Fluid Mech., 14 (2012), 33-41

    2012

  41. [41]

    Miyakawa and M

    T. Miyakawa and M. E. Schonbek , On optimal decay rates for weak solutions to the Navier-Stokes equations in R ^n , Math. Bohem. 126 (2001), no. 2, 443--455

    2001

  42. [42]

    Nguyen, A

    H. Nguyen, A. Gibbs, F. Healy, O. Shindell, R. Cortez, K. Brown, J. McCoy and B. Rodenborn , Using theory and experiments of spheres moving near boundaries to optimize the method of images for regularized Stokeslets, Phys. Rev. Fluids 10 (2025), 033101

    2025

  43. [43]

    E. M. Ouhabaz , Analysis of heat equations on domains, vol. 31 of London Mathematical Society Monographs Series, Princeton University Press, Princeton, NJ, 2005

    2005

  44. [44]

    A. F. Pillow and R. Paull , Conically similar viscous flows. Part 1. Basic conservation principles and characterization of axial causes in swirl-free flow. J. Fluid Mech. 155, 327–341 (1985)

    1985

  45. [45]

    A. F. Pillow and R. Paull , Conically similar viscous flows. Part 2. One-parameter swirl-free flows. J. Fluid Mech. 155, 343–358 (1985)

    1985

  46. [46]

    A. F. Pillow and R. Paull , Conically similar viscous flows. Part 3. Characterization of axial causes in swirling flow and the one-parameter flow generated by a uniform half-line source of kinematic swirl angular momentum. J. Fluid Mech. 155, 359–379 (1985)

    1985

  47. [47]

    Rajamanickam and A

    P. Rajamanickam and A. D. Weiss , A note on viscous flow induced by half-line sources bounded by conical surfaces. Quart. J. Mech. Appli. Math., 73 (2020), no. 1, 24-35

    2020

  48. [48]

    H. L. Royden , Real analysis. (1988) Krishna Prakashan Media, no. 6

    1988

  49. [49]

    M. E. Schonbek, L^2 decay for weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 88 (1985), no. 3, 209--222

    1985

  50. [50]

    M. E. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations 11 (1986), no. 7, 733--763

    1986

  51. [51]

    Serrin , The swirling vortex

    J. Serrin , The swirling vortex. Philos. Trans. Roy. Soc. London Ser. A, 271 (1972), no. 1214, 325-360

    1972

  52. [52]

    N. A. Slezkin , On an integrability case of full differential equations of the motion of a viscous fluid. Fluchen. Zapiski Moskov. Gosud. Universiteta, 2 (1934), 89-90

    1934

  53. [53]

    auser Advanced Texts Balser Lehrb\

    H. Sohr , The Navier-Stokes Equations: An Elementary Functional Analytic Approach. Birkh\"auser Advanced Texts Balser Lehrb\"urcher, Birkh\"auser Basel, November 2013

    2013

  54. [54]

    H. B. Squire , The round laminar jet. Quart. J. Mech. Appl. Math., 4 (1951), no. 3, 321-329

    1951

  55. [55]

    H. B. Squire , XCI. Some viscous fluid flow problems I: Jet emerging from a hole in a plane wall. The London, Edinburgh and Dublin Phil. Mag. and J. of Sci., 43 (1952), no. 344, 942-945

    1952

  56. [56]

    S ver\`ak , On Landau's solutions of the Navier-Stokes equations

    V. S ver\`ak , On Landau's solutions of the Navier-Stokes equations. J. Math. Sci., 179 (2011), no. 1, 208-229

    2011

  57. [57]

    Taniuchi , On uniqueness of mild L^ 3, -solutions on the whole time axis to the N avier-- S tokes equations in unbounded domains , Math

    Y. Taniuchi , On uniqueness of mild L^ 3, -solutions on the whole time axis to the N avier-- S tokes equations in unbounded domains , Math. Ann., 389 (2024), pp. 2561--2594

    2024

  58. [58]

    Tsai , Lectures on Navier-Stokes Equations

    T.-P. Tsai , Lectures on Navier-Stokes Equations. Graduate Studies in Mathematics, 192. American Mathematical Society, Providence, RI, 2018

    2018

  59. [59]

    C. Y. Wang , Effect of spreading of material on the surface of a fluid—an exact solution. Internat. J. Non-Linear Mech., 6 (1971), no. 6, 255-262

    1971

  60. [60]

    Yamazaki , The N avier-- S tokes equations in the weak- L^n space with time-dependent external force , Math

    M. Yamazaki , The N avier-- S tokes equations in the weak- L^n space with time-dependent external force , Math. Ann., 317 (2000), pp. 635--675

    2000

  61. [61]

    V. I. Yatseyev , On a class of exact solutions of the equations of motion of a viscous fluid. (1953) NACA-TM-1349

    1953

This paper was first reviewed by grok-4.3 on June 26, 2026.