Vanishing viscosity limit to vortex sheet for the isentropic compressible circularly symmetric 2D flow

Helong Lu

arxiv: 1906.10599 · v1 · pith:AABAG2ZUnew · submitted 2019-06-25 · 🧮 math.AP

Vanishing viscosity limit to vortex sheet for the isentropic compressible circularly symmetric 2D flow

Helong Lu This is my paper

Pith reviewed 2026-05-25 16:27 UTC · model grok-4.3

classification 🧮 math.AP

keywords vanishing viscosity limitvortex sheetisentropic compressible flowNavier-Stokes equationsboundary layerLagrangian coordinatescircular symmetry

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The pith

The isentropic compressible viscous flow approximates the inviscid vortex sheet away from the boundary and contact discontinuity, with a vortex layer only for angular velocity near the boundary.

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

The paper establishes that the solutions of the isentropic compressible Navier-Stokes equations approach those of the Euler equations as viscosity goes to zero. This approximation holds in the L^∞ norm away from the boundary and the contact discontinuity. Near the boundary and discontinuity, a boundary layer appears in the angular velocity but not in the radial velocity or pressure. Sympathetic readers would care because it provides a rigorous justification for using inviscid models in low-viscosity compressible flows while pinpointing the locations where viscous effects persist.

Core claim

Away from the boundary and the contact discontinuous the isentropic compressible viscous flow can be approximated by the corresponding inviscid flow, near the boundary (the contact discontinuous) there is a boundary layer (vortex layer) for the angular velocity in the leading order expansion of solution, while the radial velocity and the pressure do not have boundary layers (vortex layers) in the leading order. The asymptotic behavior of solutions is rigorously justified in the L^∞ space for the small viscosities limit in the Lagrangian coordinates.

What carries the argument

Vanishing viscosity limit in Lagrangian coordinates for circularly symmetric isentropic compressible 2D flow, revealing a selective vortex layer in angular velocity.

If this is right

The viscous solution converges to the inviscid vortex sheet solution in L^∞ away from the boundary and contact discontinuity as viscosity tends to zero.
A boundary layer or vortex layer forms for the angular velocity near the boundary and contact discontinuity in the leading order.
Radial velocity and pressure do not exhibit boundary layers in the leading order expansion.
The justification applies specifically under circular symmetry and impermeable boundary conditions in the 2D exterior domain.

Where Pith is reading between the lines

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

This indicates that viscosity effects in compressible vortex sheets are component-selective, affecting rotation more than radial motion.
The Lagrangian formulation used here may allow similar analyses for other symmetric compressible flow problems.
Numerical validation could involve comparing high-resolution simulations of viscous and inviscid cases to measure layer thickness.

Load-bearing premise

The corresponding Euler equations possess vortex sheet solutions under the stated circular symmetry, impermeable boundary conditions, and 2D exterior domain geometry.

What would settle it

If numerical or analytical computations show that the L^∞ difference between viscous and inviscid solutions remains bounded away from zero outside the boundary region as viscosity approaches zero, the claimed limit would be disproved.

read the original abstract

In this paper, we consider the small viscosity limit problem for the isentropic compressible Navier-Stokes equations in a 2D exterior domain with impermeable boundary conditions , and the corresponding Euler equations have vortex sheet solutions.We obtain that away from the boundary and the contact discontinuous the isentropic compressible viscous flow can be approximated by the corresponding inviscid flow, near the boundary (the contact discontinuous) there is a boundary layer (vortex layer)for the angular velocity in the leading order expansion of solution, while the radial velocity and the pressure do not have boundary layers (vortex layers) in the leading order. We rigorously justify the asymptotic behavior of solutions in the $L^{\infty}$ space for the small viscosities limit in the Lagrangian coordinates.

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 offers a detailed asymptotic description of selective boundary layers in the compressible vortex sheet limit under circular symmetry, but the justification rests on an assumed existence result for the inviscid solution.

read the letter

Here's the situation with this paper. It looks at the small viscosity limit for isentropic compressible Navier-Stokes in a 2D exterior domain with impermeable boundaries, under circular symmetry. The claim is that away from the boundary and the discontinuity, the viscous flow approximates the inviscid one, but near the contact discontinuity there's a vortex layer only in the angular velocity at leading order. Radial velocity and pressure don't show that. They say they justify the asymptotic behavior rigorously in L^∞ in Lagrangian coordinates. What the paper does is give a concrete picture of the layer structure in this symmetric compressible case. That's new in the sense that it specifies the selective presence of the layer, which isn't obvious a priori. The use of Lagrangian coordinates for the justification is a reasonable choice for tracking the discontinuity. On the soft spots, the result depends on the Euler equations having vortex sheet solutions in this setting. The abstract treats that as known, but doesn't derive or reference a specific existence result. If that existence isn't established independently, the limit statement is only as good as that assumption. The abstract doesn't include any estimates or proof sketch, so the rigor of the L^∞ justification can't be checked here. That makes the soundness provisional. No signs of circular reasoning or invented entities. The work seems focused on the asymptotic analysis. This paper is for specialists in compressible fluid dynamics and boundary layer problems. Someone looking for examples of how layers behave differently for different velocity components in symmetric flows could get something out of it. It has enough of a specific result to warrant a serious referee, even if revisions are needed to firm up the existence part. Recommendation: It should go to peer review rather than desk reject, provided the full manuscript delivers on the estimates.

Referee Report

2 major / 0 minor

Summary. The manuscript considers the vanishing-viscosity limit for the isentropic compressible Navier-Stokes equations in a 2D exterior domain with impermeable boundary conditions and circular symmetry. It claims that, away from the boundary and contact discontinuity, the viscous flow approximates the corresponding inviscid Euler flow; near the boundary (contact discontinuity) a boundary layer (vortex layer) appears for the angular velocity in the leading-order expansion, while the radial velocity and pressure have no such layers. The asymptotic behavior is asserted to be justified rigorously in the L^∞ topology in Lagrangian coordinates, under the premise that the Euler equations possess vortex-sheet solutions.

Significance. If the result holds, it would supply a rigorous L^∞ justification of the vanishing-viscosity limit to a vortex sheet for a compressible system under symmetry and boundary conditions, together with a selective description of the leading-order boundary layer. Such a result could be useful for singular-limit analysis in fluid dynamics. The strength of the claim, however, is conditional on the existence of the target discontinuous Euler solution, which is invoked but not constructed or referenced.

major comments (2)

[Abstract] Abstract: the statement that “the corresponding Euler equations have vortex sheet solutions” is invoked without derivation, citation to a well-posedness result, or construction for the discontinuous solutions under the stated circular symmetry, impermeable boundary conditions, and 2D exterior domain geometry. This existence is load-bearing for the limit claim.
[Abstract] No error bounds, convergence rates, or outline of the L^∞ estimates in Lagrangian coordinates appear in the abstract or summary material; without these the soundness of the rigorous justification cannot be evaluated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major comments point by point below.

read point-by-point responses

Referee: [Abstract] Abstract: the statement that “the corresponding Euler equations have vortex sheet solutions” is invoked without derivation, citation to a well-posedness result, or construction for the discontinuous solutions under the stated circular symmetry, impermeable boundary conditions, and 2D exterior domain geometry. This existence is load-bearing for the limit claim.

Authors: We agree that the existence of vortex sheet solutions to the Euler system is an assumption rather than a result established in the manuscript. The paper's contribution is the vanishing-viscosity analysis conditional on the existence of such solutions under the given symmetry and boundary conditions. We will revise the abstract and introduction to state explicitly that the limit holds assuming the corresponding Euler equations admit vortex sheet solutions, and we will note that the construction of these discontinuous solutions lies outside the scope of the present work. revision: yes
Referee: [Abstract] No error bounds, convergence rates, or outline of the L^∞ estimates in Lagrangian coordinates appear in the abstract or summary material; without these the soundness of the rigorous justification cannot be evaluated.

Authors: The abstract is a high-level summary. The L^∞ convergence statement and the Lagrangian-coordinate estimates are contained in the main theorems and proofs. We will add a short clause to the abstract indicating that the approximation is justified in the L^∞ topology in Lagrangian coordinates. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation is self-contained asymptotic analysis.

full rationale

The paper conducts a direct rigorous justification of the vanishing-viscosity limit for the isentropic compressible Navier-Stokes system to a pre-assumed vortex-sheet solution of the corresponding Euler equations, under the stated symmetry and boundary conditions. The existence of the target Euler solution is invoked as an external hypothesis rather than derived or fitted within the paper; the L^∞ estimates and boundary-layer analysis in Lagrangian coordinates proceed from the NS equations and do not reduce to any self-referential input, self-citation chain, or constructed prediction. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the existence of vortex-sheet solutions to the Euler system and on the circular symmetry plus impermeable boundary setup; these are standard domain assumptions in the field rather than new postulates.

axioms (1)

domain assumption The Euler equations admit vortex sheet solutions in the circularly symmetric 2D exterior domain with impermeable boundaries.
Invoked directly in the abstract as the inviscid target.

pith-pipeline@v0.9.0 · 5650 in / 1274 out tokens · 31913 ms · 2026-05-25T16:27:55.519373+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

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F.Rousset , Stability of small amplitude boundary laye rs for mixed hyperbolic-parabolic systems, Trans. Amer. Math. Soc., 355 (2003), pp. 2991 – 3008

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Sammartino, R.E

M. Sammartino, R.E. Caﬂish , Zero viscosity limit for an alytic solutions of the Navier-Stokes equation on a half-space, I. existence for Euler and Prandtl equation s, Comm. Math. Phys., 192 (1998), pp. 433 – 461

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Sammartino , R.E

M. Sammartino , R.E. Caﬂish , Zero viscosity limit for an alytic solutions of the Navier-Stokes equation on a half-space, II. construction of the Navier-Stokes solu tions, Comm. Math. Phys., 192 (1998), pp. 463 – 491

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D. Serre, K. Zumbrun , Boundary layers stability in real vanishing viscosity limit, Comm. Math. Phys., 221 (2001), pp. 267 – 292

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Temam, X.M

M. Temam, X.M. W ang , boundary layers associated with in compressible Navier-Stokes equations: The noncharacteristic boundary case, J. Diﬀential Equations, 179 (2002), pp. 647 – 686

work page 2002
[27]

W ang, M

Y.G. W ang, M. Williams, The inviscid limit and stabilit y of characteristic boundary layer for the com- pressible Navier-Stokes equations with Navier-friction b oundary condtions, Ann.Inst. Fourier (Grenble), 62 (2012), pp. 2257 – 2314

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W ang, Z.P

Y.G. W ang, Z.P. Xin , Zero-viscosity limit of the linear ized compressible Navier-Stokes equations with highly oscillatory forces in the half-plane, SIAM. J. Math. Anal., 37 (2006), pp. 1256 – 1298

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Xin , Zero dissipation limit to rarefaction waves f or the one-dimensional NavierCStokes equations of compressible isentropic gases, Comm

Z.P. Xin , Zero dissipation limit to rarefaction waves f or the one-dimensional NavierCStokes equations of compressible isentropic gases, Comm. Pure Appl. Math., 46 (5) (1993), pp. 621 – 665

work page 1993
[30]

Z.P. Xin, T. Yanagisawa , Zero-viscosity limit of the li nearized Navier-Stokes equations for a comprssible viscous ﬂuid in the half-plane, Comm. Pure Appl. Math., 52 (1999), pp. 479 – 541

work page 1999
[31]

Z.P. Xin, L. Zhang , On the global existence of solutions to the Prandtl’s system, Adv. Math., 181 (2004), pp 88 – 133

work page 2004
[32]

S.H.Yu, Zero-dissipation limit of solution with shock s for systems of hyperbolic conservation laws, Arch. Ration. Mech. Anal., 146 (4) (1999), pp. 275 – 370

work page 1999
[33]

Zeng , Stability of a superposition of shock waves w ith contact discontinuities for systems of viscous conservation laws , J

H.H. Zeng , Stability of a superposition of shock waves w ith contact discontinuities for systems of viscous conservation laws , J. Diﬀerential Equations 246 (2009), pp. 2081 – 2102. 31 32 HELONG LU,

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[34]

Zhang, R.H

Y.H. Zhang, R.H. Pan, Y. W ang, Z. Tan , Zero dissipation l imit with two interacting shocks of the 1D non-isentropic Navier-Stokes equation, Indiana Univ. Mat h. J., 62 (1) (2013), pp. 249 –309. E-mail address : luhelong1988@126.com,

work page 2013

[1] [1]

W ang, C.J

R.Alexander,Y.G. W ang, C.J. Xu, T,Yang , W ell-posednes s of the Prandtl equation in sobolev spaces, J. Amer. Math. Soc., 28 (2015), pp. 745–784

work page 2015

[2] [2]

Corli , Asymptotic analysis of contact discontinuiti es, Ann

A. Corli , Asymptotic analysis of contact discontinuiti es, Ann. Mat. Pura Appl. 173 (4) (1997), pp. 163–202

work page 1997

[3] [3]

Pure Appl

W.E, B.Engquist , Blow up of solutions of the unsteady Pra ndtl’s equation, Comm. Pure Appl. Math., 50 (1997), pp. 1287– 1293

work page 1997

[4] [4]

Dormy , On the ill-posedness of the P randtl equation, J

D.G´ erard-Varet, E. Dormy , On the ill-posedness of the P randtl equation, J. Amer. Math. Soc., 23 (2010), pp. 591– 609

work page 2010

[5] [5]

Goodman, Z.P

J. Goodman, Z.P. Xin , Viscosity limits for piecewise smo oth solutions to systems of conservation laws, Arch. Ration. Mech. Anal., 121 (3) (1992), pp. 235 – 265. 30

work page 1992

[6] [6]

Grenier , On the nonlinear instability of Euler and Pra ndtl equations, Comm

E. Grenier , On the nonlinear instability of Euler and Pra ndtl equations, Comm. Pure Appl. Math., 53 (2000), pp. 1067– 1091

work page 2000

[7] [7]

Y. Guo, T. Nguyen , A note on the Prandtl boundary layers Co mm. Pure Appl. Math., 64 (2011), pp. 1416 – 1438

work page 2011

[8] [8]

Hong, J.K

L. Hong, J.K. Hunter , Singularity formation and instabi lity in the unsteady inviscid and viscous Prandtl equations, Commun. Matn. Sci., 1 (2003), pp. 293 – 316

work page 2003

[9] [9]

Huang, J

F.M. Huang, J. Li, A. Matsumura , Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navie r-Stokes system , Arch. Ration. Mech. Anal., 197 (1) (2010), pp. 89 – 116

work page 2010

[10] [10]

Huang, A

F.M. Huang, A. Matsumura, Z.P. Xin , Stability of contac t discontinuities for the 1-D compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (1) (2006), pp. 55 – 77

work page 2006

[11] [11]

Hartman, A

P. Hartman, A. Wintner On hyperbolic partial diﬀerenti al equations, Amer. J. Math., 74 (1952), pp. 834–864

work page 1952

[12] [12]

Huang, Y

F.M. Huang, Y. W ang, T. Yang , Vanishing viscosity limit of the compressible Navier-Stokes equations for solutions to a Riemann problem, Arch. Ration. Mech. Anal ., 203 (2) (2012), pp. 379 – 413

work page 2012

[13] [13]

Liu, Y.G

C.J. Liu, Y.G. W ang , stability of boundary layers for th e nonisentropic compressible circularly symmetric 2D ﬂow, SIAM. J. Math. Anal., 46 (1) (2014), pp.256 – 309

work page 2014

[14] [14]

Ma, Zero dissipation limit to strong contact disco ntinuity for the 1-D compressible NavierCStokes equations, J

S.X. Ma, Zero dissipation limit to strong contact disco ntinuity for the 1-D compressible NavierCStokes equations, J. Dﬀerential Equations , 248 (2010), pp. 95 –110

work page 2010

[15] [15]

A.Majda , Compressible ﬂuid ﬂow and systems of conserva tion laws in several space variables, Springer- Verlag, New York, 1984

work page 1984

[16] [16]

N.Masmoudi , The Euler limit of the Navier-Stokes equat ions,and rotating ﬂuids with boundarty, Arch. Ration. Maech. Anal., 142 (4) (1998), pp. 375 – 394

work page 1998

[17] [17]

Masmoudi, T.K

N. Masmoudi, T.K. W ong , Local-in-time existence and un iqueness of solutions to the Prandtl equations by energy methods , Comm. Pure Appl. Math. 68 (2015), pp. 1683–1741

work page 2015

[18] [18]

Oleinik, V.N

O.A. Oleinik, V.N. Samokhin , Mathematical Models in Bo undary Layers Theory, Chapman & Hall/ CRC, Boca Raton, FL, 1999

work page 1999

[19] [19]

L.Prandtl , ¨Uber Fl¨ ussigkeitsbewegungen bei sehr kleiner Reibung, inVerh. Int. Math. Konger.,Heideberg, Germany, (1904), pp. 484–494

work page 1904

[20] [20]

Diﬀerential Equations, 210 (2005) , pp

F.Rousset , Characteristic boundary layers in real van ishing viscosity limits, J. Diﬀerential Equations, 210 (2005) , pp. 25 – 64

work page 2005

[21] [21]

F.Rousset , Stability of small amplitude boundary laye rs for mixed hyperbolic-parabolic systems, Trans. Amer. Math. Soc., 355 (2003), pp. 2991 – 3008

work page 2003

[22] [22]

Sammartino, R.E

M. Sammartino, R.E. Caﬂish , Zero viscosity limit for an alytic solutions of the Navier-Stokes equation on a half-space, I. existence for Euler and Prandtl equation s, Comm. Math. Phys., 192 (1998), pp. 433 – 461

work page 1998

[23] [23]

Sammartino , R.E

M. Sammartino , R.E. Caﬂish , Zero viscosity limit for an alytic solutions of the Navier-Stokes equation on a half-space, II. construction of the Navier-Stokes solu tions, Comm. Math. Phys., 192 (1998), pp. 463 – 491

work page 1998

[24] [24]

Serre, K

D. Serre, K. Zumbrun , Boundary layers stability in real vanishing viscosity limit, Comm. Math. Phys., 221 (2001), pp. 267 – 292

work page 2001

[25] [25]

M.E.Taylor , Partial Diﬀerential Equations III, Nonli near Equations, Springer-Verlag, New York, 1996

work page 1996

[26] [26]

Temam, X.M

M. Temam, X.M. W ang , boundary layers associated with in compressible Navier-Stokes equations: The noncharacteristic boundary case, J. Diﬀential Equations, 179 (2002), pp. 647 – 686

work page 2002

[27] [27]

W ang, M

Y.G. W ang, M. Williams, The inviscid limit and stabilit y of characteristic boundary layer for the com- pressible Navier-Stokes equations with Navier-friction b oundary condtions, Ann.Inst. Fourier (Grenble), 62 (2012), pp. 2257 – 2314

work page 2012

[28] [28]

W ang, Z.P

Y.G. W ang, Z.P. Xin , Zero-viscosity limit of the linear ized compressible Navier-Stokes equations with highly oscillatory forces in the half-plane, SIAM. J. Math. Anal., 37 (2006), pp. 1256 – 1298

work page 2006

[29] [29]

Xin , Zero dissipation limit to rarefaction waves f or the one-dimensional NavierCStokes equations of compressible isentropic gases, Comm

Z.P. Xin , Zero dissipation limit to rarefaction waves f or the one-dimensional NavierCStokes equations of compressible isentropic gases, Comm. Pure Appl. Math., 46 (5) (1993), pp. 621 – 665

work page 1993

[30] [30]

Z.P. Xin, T. Yanagisawa , Zero-viscosity limit of the li nearized Navier-Stokes equations for a comprssible viscous ﬂuid in the half-plane, Comm. Pure Appl. Math., 52 (1999), pp. 479 – 541

work page 1999

[31] [31]

Z.P. Xin, L. Zhang , On the global existence of solutions to the Prandtl’s system, Adv. Math., 181 (2004), pp 88 – 133

work page 2004

[32] [32]

S.H.Yu, Zero-dissipation limit of solution with shock s for systems of hyperbolic conservation laws, Arch. Ration. Mech. Anal., 146 (4) (1999), pp. 275 – 370

work page 1999

[33] [33]

Zeng , Stability of a superposition of shock waves w ith contact discontinuities for systems of viscous conservation laws , J

H.H. Zeng , Stability of a superposition of shock waves w ith contact discontinuities for systems of viscous conservation laws , J. Diﬀerential Equations 246 (2009), pp. 2081 – 2102. 31 32 HELONG LU,

work page 2009

[34] [34]

Zhang, R.H

Y.H. Zhang, R.H. Pan, Y. W ang, Z. Tan , Zero dissipation l imit with two interacting shocks of the 1D non-isentropic Navier-Stokes equation, Indiana Univ. Mat h. J., 62 (1) (2013), pp. 249 –309. E-mail address : luhelong1988@126.com,

work page 2013