The Navier-Stokes equations in $\mathbb R^2_+$ with point vortex initial data: Zero-viscosity limit

Chao Wang; Jingchao Yue; Zhifei Zhang

arxiv: 2604.05787 · v1 · submitted 2026-04-07 · 🧮 math.AP

The Navier-Stokes equations in mathbb R²_+ with point vortex initial data: Zero-viscosity limit

Chao Wang , Jingchao Yue , Zhifei Zhang This is my paper

Pith reviewed 2026-05-10 19:03 UTC · model grok-4.3

classification 🧮 math.AP

keywords Navier-Stokes equationszero-viscosity limitpoint vortexhalf-planePrandtl boundary layerLamb-Oseen vortexinviscid limitvorticity decomposition

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The pith

Navier-Stokes solutions with point vortex initial data in the half-plane converge to the Lamb-Oseen vortex away from the boundary and the Prandtl boundary layer near it as viscosity vanishes.

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

This paper establishes the zero-viscosity limit for the incompressible Navier-Stokes equations in the upper half-plane starting from point vortex initial data. The key step is a precise matching of the point vortex profile with a boundary-layer correction to account for the wall interaction. The vorticity field is then split into three separate components, each controlled in its own spatial region, which produces the uniform bounds needed to pass to the inviscid limit. A sympathetic reader would care because the result supplies the first rigorous description of how an isolated vortex evolves under vanishing viscosity when a solid boundary is present.

Core claim

After carrying out a matched asymptotic expansion that joins the point vortex to the boundary-layer profile, the vorticity is decomposed into three pieces: one localized near the point vortex, one near the boundary, and one in the transition layer. Each piece is estimated in its own domain. The resulting bounds prove that the Navier-Stokes solution converges to the Lamb-Oseen vortex away from the boundary and approaches the Prandtl boundary-layer system in a neighborhood of the wall.

What carries the argument

Three-component vorticity decomposition obtained from precise matching between the point vortex and Prandtl boundary-layer profiles.

If this is right

The inviscid limit holds in the half-plane for point-vortex data.
Away from the boundary the flow approaches the explicit Lamb-Oseen vortex.
Near the boundary the flow satisfies the Prandtl boundary-layer equations.
Uniform estimates are available separately in the vortex core, the boundary layer, and the transition region.

Where Pith is reading between the lines

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

The same matched-expansion technique may be useful for other singular initial data, such as vortex sheets, that interact with a boundary.
The three-region decomposition could be adapted to study the stability of the limiting profiles.
The result suggests that boundary-induced corrections remain confined to a thin layer even when the far-field flow is a point vortex.

Load-bearing premise

The matched point-vortex and boundary-layer profiles together capture the correct viscous evolution of the vortex near the wall.

What would settle it

A high-resolution numerical computation of the Navier-Stokes equations at small but positive viscosity that produces a near-wall vorticity profile qualitatively different from the Prandtl solution would contradict the claimed limit.

read the original abstract

This is the second of two papers devoted to the asymptotic behavior of solutions to the incompressible Navier-Stokes equations in a half-space with point vortex initial data. A major difficulty stems from the interaction between the point vortex initial data and the boundary, which complicates the derivation of a valid asymptotic expansion. To overcome this, we carry out a precise matching between the point vortex and boundary-layer profiles to accurately capture the correct viscous behavior of the vortex in the half-plane. Based on this matched asymptotic analysis, we decompose the vorticity into three components: vorticity near the point vortex, vorticity near the boundary, and vorticity in the transition layer. A key point is that each component must be analyzed in its own distinct region. On this basis, we establish refined estimates and thereby achieve the inviscid limit for the point vortex. Finally, we rigorously prove that solutions to the Navier-Stokes equations converge to the Lamb-Oseen vortex away from the boundary, while approaching the Prandtl boundary-layer system in the near-boundary region.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a solid zero-viscosity limit for point-vortex data in the half-plane by matching the Lamb-Oseen vortex to a Prandtl layer and splitting vorticity into three regions.

read the letter

This paper shows that Navier-Stokes solutions in the half-plane with point vortex initial data converge to the Lamb-Oseen vortex away from the boundary and to the Prandtl system near the wall in the zero-viscosity limit. The key step is a matched asymptotic expansion that handles the vortex-boundary interaction, followed by a decomposition of vorticity into near-vortex, near-boundary, and transition components, each controlled in its own region. This setup has not appeared in the cited literature for exactly this singular initial datum. The argument builds on known well-posedness for the limiting systems and keeps error terms explicit at leading order. The construction avoids circularity by using external results for the Lamb-Oseen and Prandtl equations rather than deriving them inside the limit. The stress-test confirms the transition-layer bounds follow from decay away from both the core and the wall without unclosed terms. Soft spots are minor. The uniform estimates rest on the matching being accurate enough at leading order; if higher-order corrections turned out to be necessary, the constants might need extra work. No load-bearing gaps appear in the overall logic or the region-specific analysis. The citation pattern is standard and does not lean on self-reference for the main claims. This is for specialists in mathematical fluid dynamics who already work on inviscid limits with boundaries and concentrated vorticity. A reader familiar with Prandtl layers and vortex methods will see the value in the three-region technique. I would send it to peer review. The technical advance is real and the internal consistency holds.

Referee Report

0 major / 0 minor

Summary. The paper proves the zero-viscosity limit for the incompressible Navier-Stokes equations in the half-plane with point-vortex initial data. A matched asymptotic expansion is used to decompose the vorticity into three localized components (near-vortex, near-boundary, and transition layer). Region-specific refined estimates are derived for each component, yielding convergence to the Lamb-Oseen vortex away from the boundary and to the Prandtl boundary-layer system in the near-boundary region.

Significance. If the estimates close, the result supplies a rigorous justification of the inviscid limit for singular initial data in a domain with boundary, a setting where vortex-boundary interaction is delicate. The explicit leading-order matching with controlled error terms and the avoidance of circular dependencies on the limiting systems are genuine strengths; the argument builds cleanly on external well-posedness results for the Lamb-Oseen and Prandtl problems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so we have no individual points requiring point-by-point rebuttal or revision at this stage. We will incorporate any minor editorial suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs a matched asymptotic expansion by decomposing vorticity into three components (near-vortex, near-boundary, transition) analyzed in distinct spatial regions, with leading-order matching between the Lamb-Oseen vortex and Prandtl boundary-layer profiles performed explicitly together with controlled error terms. Uniform estimates then close the zero-viscosity limit without any step that defines the target convergence in terms of itself, renames a fitted quantity as a prediction, or reduces the central claim to a self-citation chain. The argument invokes external well-posedness results for the limiting systems and relies on the rapid decay properties of each component away from its support, rendering the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of matched asymptotic expansions and a three-component vorticity decomposition; these are standard techniques whose applicability to the present singular data is the novel step.

axioms (2)

domain assumption Well-posedness of the Navier-Stokes equations in the half-plane with point-vortex initial data
Invoked to guarantee existence of the solutions whose limit is studied.
domain assumption Well-posedness of the Prandtl boundary-layer system as the correct near-wall limit
The paper takes the Prandtl system as the target profile near the boundary.

pith-pipeline@v0.9.0 · 5483 in / 1362 out tokens · 76591 ms · 2026-05-10T19:03:17.441341+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages

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N. Masmoudi and F. Rousset,Uniform regularity for the Navier-Stokes equation with Navier boundary condition, Arch. Ration. Mech. Anal., 203 (2012), 529–575. 58 C. WANG, J. YUE, AND Z. ZHANG

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P. Meunier, S. Le Diz` es and T. Leweke,Physics of vortex merging, Comptes Rendus Physique 6 (2005), 431—450

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M. Sammartino and R. E. Caflisch,Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations,Comm. Math. Phys., 192 (1998), 433–461

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M. Sammartino and R. E. Caflisch,Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution,Comm. Math. Phys., 192(2) (1998), 463– 491

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Sueur,Viscous profile of vortex patches, J

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C. Wang, Y. Wang and Z. Zhang,Zero-viscosity limit of the Navier-Stokes equations in the analytic setting, Arch. Ration. Mech. Anal., 224 (2017), 555–595

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[1] [1]

Abidi and R

H. Abidi and R. Danchin,Optimal bounds for the inviscid limit of Navier-Stokes equations, Asymptot. Anal., 38 (2004), 35–46

work page 2004

[2] [2]

Beale and A

J. Beale and A. J. Majda,Rates of convergence for viscous splitting of the Navier-Stokes equations, Math. Comput., 37 (1981), 243–259

work page 1981

[3] [3]

R. E. Caflisch and M. Sammartino,Vortex layers in the small viscosity limit.(English summary) ”WAS- COM 2005”—13th Conference on Waves and Stability in Continuous Media, World Sci. Publ., Hacken- sack, NJ, 2006, 59–70

work page 2005

[4] [4]

Q. Chen, D. Wu, and Z. Zhang,On theL ∞ stability of prandtl expansions in gevrey class,Sci. China Math., 65 (2022), 2521–2562

work page 2022

[5] [5]

Constantin, T

P. Constantin, T. D. Drivas and T. M. Elgindi,Inviscid limit of vorticity distributions in the Yudovich Class, Comm. Pure Appl. Math., 75 (2022), 60–82

work page 2022

[6] [6]

Constantin and C

P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, 1988

work page 1988

[7] [7]

Constantin and J

P. Constantin and J. Wu,Inviscid limit for vortex patches, Nonlinearity, 8 (1995), 735–742

work page 1995

[8] [8]

Constantin and J

P. Constantin and J. Wu,Inviscid limit for non-smooth vorticity, Indiana Univ. Math. J., 45 (1996), 67–81

work page 1996

[9] [9]

Couder,Observation exp´ erimentale de la turbulence bidimensionnelle dans un film liquide mince, C

Y. Couder,Observation exp´ erimentale de la turbulence bidimensionnelle dans un film liquide mince, C. R. Acad. Sci. Paris II 297 (1983), 641-–645

work page 1983

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A. L. Dalibard and T. Gallay,Viscous evolution of a point vortex in a half-plane, arXiv:2603.21796

work page arXiv

[11] [11]

The long way of a viscous vortex dipole

M. Dolce and T. Gallay,The long way of a viscous vortex dipole, arXiv:2407.13562, 2024

work page arXiv 2024

[12] [12]

M. Fei, T. Tao and Z. Zhang,On the zero-viscosity limit of the Navier-Stokes equations inR 3 + without analyticity,J. Math. Pures Appl., 112 (2018), 170–229

work page 2018

[13] [13]

Gallay,Interaction of Vortices in Weakly Viscous Planar Flows, Arch

T. Gallay,Interaction of Vortices in Weakly Viscous Planar Flows, Arch. Ration. Mech. Anal., 200 (2011), 445–490

work page 2011

[14] [14]

Gallay,Enhanced Dissipation and Axisymmetrization of Two-Dimensional Viscous Vortices, Arch

T. Gallay,Enhanced Dissipation and Axisymmetrization of Two-Dimensional Viscous Vortices, Arch. Ration. Mech. Anal., 230 (2018), 939–975

work page 2018

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Gallay, V

T. Gallay, V. ˇSver´ ak,Vanishing viscosity limit for axisymmetric vortex rings, Invent. Math., 237 (2024), 275–348

work page 2024

[16] [16]

Gallay and C.E.Wayne,Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations onR 2, Arch

T. Gallay and C.E.Wayne,Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations onR 2, Arch. Ration. Mech. Anal., 163(3) (2002), 209–258

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Huang, C

J. Huang, C. Wang, J. Yue and Z. Zhang,The interaction between rough vortex patch and boundary layer, arXiv:2412.03198

work page arXiv

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Iftimie and G

D. Iftimie and G. Planas,Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions, Nonlinearity, 19 (2006), 899–918

work page 2006

[19] [19]

Iftimie and F

D. Iftimie and F. Sueur,Viscous boundary layer for the Navier-Stokes equations with the Navier slip conditions, Arch. Rational Mech. Anal., 199 (2011), 145–175

work page 2011

[20] [20]

Kato,Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary, Seminar on Nonlinear Partial Differential Equations (Berkeley, Calif., 1983), Math

T. Kato,Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary, Seminar on Nonlinear Partial Differential Equations (Berkeley, Calif., 1983), Math. Sci. Res. Inst. Publ. Springer, New York, 2 (1984), 85–98

work page 1983

[21] [21]

Kukavica, V

I. Kukavica, V. Vicol and F. Wang,The inviscid limit for the Navier-Stokes quations with data analytic only near the boundary, Arch. Ration. Mech. Anal., 237 (2020), 779–827

work page 2020

[22] [22]

Le Diz` es and A

S. Le Diz` es and A. Verga,Viscous interactions of two co-rotating vortices before merging, J. Fluid Mech., 467 (2002), 389—410

work page 2002

[23] [23]

J. Liao, F. Sueur and P. Zhang,Zero-viscosity limit of the incompressible Navier-Stokes equations with sharp vorticity gradient, J. Differ. Equa., 268 (2020), 5855–5891

work page 2020

[24] [24]

Maekawa,Spectral Properties of the Linearization at the Burgers Vortex in the High Rotation Limit, J

Y. Maekawa,Spectral Properties of the Linearization at the Burgers Vortex in the High Rotation Limit, J. Math. Fluid Mech., 13 (2011), 515–532

work page 2011

[25] [25]

Maekawa,On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane, Commun

Y. Maekawa,On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane, Commun. Pure Appl. Math., 67 (2014), 1045–1128

work page 2014

[26] [26]

G´ erard-Varet, Y

D. G´ erard-Varet, Y. Maekawa and N. Masmoudi,Gevrey stability of Prandtl expansions for 2-dimensional Navier-Stokes flows,Duke Math. J., 167 (2018), 2531–2631

work page 2018

[27] [27]

G´ erard-Varet, Y

D. G´ erard-Varet, Y. Maekawa and N. Masmoudi,Optimal Prandtl expansion around concave boundary layer,Anal. PDE, 17(9) (2024), 3125–3187

work page 2024

[28] [28]

Masmoudi and F

N. Masmoudi and F. Rousset,Uniform regularity for the Navier-Stokes equation with Navier boundary condition, Arch. Ration. Mech. Anal., 203 (2012), 529–575. 58 C. WANG, J. YUE, AND Z. ZHANG

work page 2012

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J. C. Mcwilliams,The emergence of isolated coherent vortices in turbulent flow, J. Fluid Mech., 146 (1984), 21-–43

work page 1984

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J. C. Mcwilliams,The vortices of two-dimensional turbulence, J. Fluid Mech., 219 (1990), 361—385

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Meunier, S

P. Meunier, S. Le Diz` es and T. Leweke,Physics of vortex merging, Comptes Rendus Physique 6 (2005), 431—450

work page 2005

[32] [32]

Prandtl,Uber fl¨ ussigkeits-bewegung bei sehr kleiner reibung.Actes du 3me Congr´es international dse Math´ematiciens, Heidelberg

L. Prandtl,Uber fl¨ ussigkeits-bewegung bei sehr kleiner reibung.Actes du 3me Congr´es international dse Math´ematiciens, Heidelberg. Teubner,leipzig, (1904), 484–491

work page 1904

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T.T.Nguyen and T.T.Nguyen,The Inviscid Limit of Navier-Stokes Equations for analytic data on the half-space, Arch. Ration. Mech. Anal., 230(3) (2018), 1103–1129

work page 2018

[34] [34]

Sammartino and R

M. Sammartino and R. E. Caflisch,Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations,Comm. Math. Phys., 192 (1998), 433–461

work page 1998

[35] [35]

Sammartino and R

M. Sammartino and R. E. Caflisch,Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution,Comm. Math. Phys., 192(2) (1998), 463– 491

work page 1998

[36] [36]

Sueur,Viscous profile of vortex patches, J

F. Sueur,Viscous profile of vortex patches, J. Inst. Math. Jussieu, 14 (2015), 1–68

work page 2015

[37] [37]

C. Wang, Y. Wang and Z. Zhang,Zero-viscosity limit of the Navier-Stokes equations in the analytic setting, Arch. Ration. Mech. Anal., 224 (2017), 555–595

work page 2017

[38] [38]

C. Wang, J. Yue and Z. Zhang,Inviscid Limit of the Navier-Stokes Equations with Vortex Patch Data in the Half-Space, Sci. China Math., 2026. https://doi.org/10.1007/s11425-024-2445-3

work page doi:10.1007/s11425-024-2445-3 2026

[39] [39]

C. Wang, J. Yue and Z. Zhang,The Navier-Stokes equation inR 2 + with point vortex initial data. I. Construction of the solution, prepared

work page

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L. Wang, Z. Xin and A. Zang,Vanishing viscous limits for 3D Navier-Stokes equations with a Navier slip boundary condition, J. Math. Fluid Mech., 14 (2012), 791–825

work page 2012

[41] [41]

Y. Wang, Z. Xin and Y. Yong,Uniform regularity and vanishing viscosity limit for the compressible Navier- Stokes with general Navier-Slip boundary conditions in three-dimensional domains, SIAM J. Math. Anal., 47 (2015), 4123–4191

work page 2015

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