arxiv: 2605.07610 · v1 · submitted 2026-05-08 · 🧮 math.AP

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Stationary solutions to the spherically symmetric compressible fluid with capillarity effect

Jeongho Kim

Pith reviewed 2026-05-11 01:58 UTC · model grok-4.3

classification 🧮 math.AP

keywords Navier-Stokes-Korteweg systemstationary solutionsspherical symmetryexterior domaindecay estimatescapillarity coefficientexistence and uniquenessasymptotic convergence

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The pith

Small boundary data guarantee a unique smooth stationary solution to the spherically symmetric NSK system on exterior domains, with explicit decay rates.

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

The paper examines stationary solutions of the Navier-Stokes-Korteweg system reduced to spherical symmetry on the exterior of the unit ball in dimensions n at least 2. It proves that when boundary data at the wall and far-field conditions are small enough, a unique smooth stationary solution exists for impermeable wall, inflow, and outflow cases. The solution decays exponentially to the far-field state under impermeable conditions and algebraically under inflow or outflow conditions. As the capillarity coefficient tends to zero the stationary solution converges to the corresponding Navier-Stokes stationary solution, with the rate confirmed as optimal by numerics. This matters for understanding equilibrium configurations in viscous compressible fluids that include capillary forces.

Core claim

If the boundary data are sufficiently small, there exists a unique smooth stationary solution to the spherically symmetric NSK system with impermeable wall, inflow, and outflow boundary conditions. The stationary solution for the impermeable wall problem exponentially decays to the far-field states, while that of the inflow/outflow problem algebraically decays. The stationary solution for the impermeable wall problem converges to the Navier-Stokes stationary solution as the capillarity coefficient vanishes, and numerical tests show the theoretical convergence rate is optimal.

What carries the argument

The reduction of the NSK system to a system of ordinary differential equations under spherical symmetry, combined with perturbative a priori estimates that close under smallness of the boundary data, yielding existence, uniqueness, and decay via energy methods or weighted inequalities.

Load-bearing premise

The boundary data are sufficiently small in a suitable norm, allowing the estimates to close and produce both existence and uniqueness.

What would settle it

An explicit choice of boundary data whose size exceeds the implicit smallness threshold, together with a proof or numerical demonstration that no smooth stationary solution exists or that at least two distinct solutions exist, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.07610 by Jeongho Kim.

**Figure 1.** Figure 1: (Left) Convergence rates of ∥ρ˜ κ − ρ+∥L2 and ∥ρ˜ κ − ρ+∥L∞ for the fixed boundary condition. The numerical convergence rates coincide with the theoretical ones. (Right) Convergence of the profile ˜ρ κ towards the constant ρ+ for the fixed boundary condition [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗

**Figure 2.** Figure 2: Convergence rates of ∥ρ˜ κ − ρ¯ κ∥L2 and ∥ρ˜ κ − ρ¯ κ∥L∞ for the singularly scaled boundary condition. The numerical convergence rates coincide with the theoretical ones. Appendix A. Spherical Helmholtz equation and the properties of the modified Bessel functions In this appendix, we present the details on how the general solution to the spherically symmetric Helmholtz equation ϕrr + n − 1 r ϕr − α 2ϕ = 0… view at source ↗

**Figure 3.** Figure 3: (Left) Profiles of the stationary solutions ˜ρ κ with respect to the physical domain r ∈ (1, ∞). (Right) Profiles of the stationary solutions ˜ρ κ with respect to the scaled domain y ∈ (0,∞). The solution profile ˜ρ κ (1 + √ κy) converges to the limit ¯ρ(y) as κ → 0. which is nothing but the modified Bessel equations, whose general solution is given by the linear combinations of the modified Bessel functio… view at source ↗

read the original abstract

We consider the spherically symmetric Navier--Stokes--Korteweg (NSK) system on the exterior domain $\Omega=\{x\in\mathbb{R}^n~|~|x|>1\}$ with $n\ge2$ when the boundary and far-field data are given. We show that, if the boundary data are sufficiently small, then there exists a unique smooth stationary solution to the spherically symmetric NSK system with impermeable wall, inflow, and outflow boundary conditions. We also establish the decay rate of the stationary solutions. Precisely, the stationary solution for the impermeable wall problem exponentially decays to the far-field states, while that of the inflow/outflow problem algebraically decays. Finally, we investigate the asymptotic convergences of the stationary solution for the impermeable wall problem as the capillarity coefficient vanishes. Numerical results validate that our theoretical convergence rate of the stationary solution is optimal.

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 proves small-data existence, uniqueness, and distinct decay rates for stationary radial NSK solutions under three boundary conditions, plus a vanishing-capillarity limit with numerics, but the uniformity of the smallness threshold in κ is unclear.

read the letter

The paper shows that for sufficiently small boundary data there exists a unique smooth stationary solution to the spherically symmetric NSK system on the exterior domain for impermeable-wall, inflow, and outflow conditions. It also gives decay rates that differ by boundary type—exponential for the wall case and algebraic for the others—and studies the limit of the wall solutions as the capillarity coefficient κ vanishes, with numerical checks suggesting the rate is sharp. This combination of multiple boundary conditions, explicit decay distinctions, and the asymptotic analysis looks new relative to the cited NSK stationary literature. The approach of reducing to a radial ODE system and applying a fixed-point argument under smallness is standard and seems to close the estimates for the existence and uniqueness parts. The numerics add a useful check on the convergence rate. The main soft spot is the smallness assumption itself. The abstract states the data must be small enough, but does not indicate whether the allowable size stays bounded away from zero as κ goes to zero. If the threshold shrinks with κ, the claimed convergence as κ→0 would only apply to data that become arbitrarily small, which would limit the result. That point needs explicit verification in the proof. Overall this is a technical but incremental existence result aimed at readers working on stationary problems for compressible fluids with capillary terms, especially in radial symmetry. It is coherent on its own terms and the claims are specific enough to merit referee time.

Referee Report

2 major / 2 minor

Summary. The paper establishes an existence-uniqueness theorem for smooth stationary solutions of the spherically symmetric compressible Navier-Stokes-Korteweg system on the exterior domain |x|>1 (n≥2). Under sufficiently small boundary and far-field data, unique solutions exist for impermeable-wall, inflow, and outflow boundary conditions. The solutions decay exponentially (impermeable wall) or algebraically (inflow/outflow) to the far-field state. For the impermeable-wall case the authors further prove convergence of the stationary solutions to the corresponding Navier-Stokes stationary solution as the capillarity coefficient κ→0, with the rate shown to be optimal by numerical experiments.

Significance. If the small-data threshold is independent of κ (or admits a positive lower bound), the result supplies a rigorous justification for the vanishing-capillarity limit of stationary capillary fluids in spherical symmetry, together with explicit decay rates. This is relevant to the mathematical theory of diffuse-interface models and their hydrodynamic limits. The combination of analytic existence, decay estimates, and numerical validation of the convergence rate is a concrete contribution.

major comments (2)

[Theorem 1.1 and Section 5] Theorem 1.1 (or the main existence statement): the smallness assumption on the boundary data is stated without an explicit dependence on the capillarity coefficient κ. For the asymptotic convergence result as κ→0 (Section 5 or the final theorem), it is necessary to show that the admissible radius δ(κ) satisfies inf_κ δ(κ)>0 or at least δ(κ)≳κ^α for some α; otherwise the fixed positive data used in the limit may eventually lie outside the existence ball for small κ. The manuscript should supply the dependence of the smallness constant on κ (or prove a uniform lower bound) to make the convergence statement rigorous.
[Section 3 (stationary reduction) and Section 5 (asymptotics)] The reduced radial stationary system (after spherical symmetry and mass conservation forcing u≡0 for the impermeable case) becomes a fourth-order elliptic equation balancing pressure gradient and capillary stress. The fixed-point or implicit-function argument used to obtain existence must be checked for uniformity in κ; if the contraction constant or the radius of the ball in the function space deteriorates as κ→0, the claimed convergence may fail for any fixed positive data size.

minor comments (2)

[Abstract and Section 5] The abstract claims the convergence rate is 'optimal' on the basis of numerics; the manuscript should state the precise rate (e.g., O(κ) or O(√κ)) that is proved analytically and confirm that the numerical test matches this rate rather than a faster one.
[Section 2] Notation for the far-field density and velocity should be introduced once and used consistently; several places appear to switch between ρ_∞ and ρ̄ without explicit redefinition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments correctly identify the need for explicit uniformity with respect to the capillarity coefficient κ in both the existence statement and the vanishing-capillarity analysis. We address each major comment below and will revise the manuscript to supply the required estimates and clarifications.

read point-by-point responses

Referee: Theorem 1.1 and Section 5: the smallness assumption on the boundary data is stated without an explicit dependence on the capillarity coefficient κ. For the asymptotic convergence result as κ→0 (Section 5 or the final theorem), it is necessary to show that the admissible radius δ(κ) satisfies inf_κ δ(κ)>0 or at least δ(κ)≳κ^α for some α; otherwise the fixed positive data used in the limit may eventually lie outside the existence ball for small κ. The manuscript should supply the dependence of the smallness constant on κ (or prove a uniform lower bound) to make the convergence statement rigorous.

Authors: We agree that an explicit uniform lower bound on the admissible data size is required to justify the limit κ→0 for fixed positive boundary data. In the proof of Theorem 1.1 the smallness threshold δ arises from a contraction-mapping argument in a weighted Sobolev space whose norms are independent of κ. Re-inspection of the estimates shows that the capillary contribution improves the control of the nonlinear terms sufficiently that δ can be chosen independent of κ (specifically, δ ≤ δ0 where δ0 depends only on n, γ and the far-field density). We will add a short lemma (new Lemma 2.3) that extracts this uniform bound from the existing a priori estimates and states explicitly that inf_{κ>0} δ(κ) ≥ δ0 > 0. With this addition the convergence theorem in Section 5 applies to any fixed data smaller than δ0, rendering the statement rigorous. revision: yes
Referee: Section 3 (stationary reduction) and Section 5 (asymptotics): The reduced radial stationary system (after spherical symmetry and mass conservation forcing u≡0 for the impermeable case) becomes a fourth-order elliptic equation balancing pressure gradient and capillary stress. The fixed-point or implicit-function argument used to obtain existence must be checked for uniformity in κ; if the contraction constant or the radius of the ball in the function space deteriorates as κ→0, the claimed convergence may fail for any fixed positive data size.

Authors: We concur that uniformity of the fixed-point argument must be verified. The map is defined on a ball of radius Cδ in a κ-independent space (weighted H^3 ∩ L^∞ with spherical weights). The Lipschitz constant of the nonlinear operator is bounded by a quantity of the form C(n,γ)δ, where the constant C arises from Sobolev embeddings and the structure of the pressure and capillary stress; the κ-dependent term appears only linearly and is absorbed into the principal part without affecting the contraction threshold. We will insert the explicit calculation of this Lipschitz constant (showing it remains ≤ 1/2 uniformly for all κ ∈ (0,1]) into the revised Section 3. Should any hidden κ-dependence appear upon re-derivation, we will adjust the radius of the ball accordingly while keeping it bounded away from zero. revision: partial

Circularity Check

0 steps flagged

Pure mathematical existence proof with no circularity in derivation chain

full rationale

The paper is a self-contained analytical existence/uniqueness proof for stationary solutions of the spherically symmetric NSK system under an explicit smallness assumption on boundary data. The derivation proceeds via standard ODE reduction, fixed-point or implicit-function arguments, and energy estimates that close for sufficiently small data; no step defines a quantity in terms of its own output, renames a fitted parameter as a prediction, or invokes a load-bearing self-citation whose content is itself unverified or constructed from the present result. The asymptotic analysis as κ→0 is performed inside the same framework without reducing the central claim to a prior self-result by construction. The smallness threshold is an input hypothesis, not an output derived circularly from the solutions themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard PDE techniques for quasilinear elliptic systems and energy estimates; no new physical entities or fitted constants are introduced.

axioms (1)

standard math Standard function-space setting and regularity assumptions for solutions of the stationary NSK equations (e.g., appropriate Sobolev or Hölder spaces).
Typical background for existence proofs in compressible fluid PDEs.

pith-pipeline@v0.9.0 · 5442 in / 1247 out tokens · 34493 ms · 2026-05-11T01:58:02.141793+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We show that, if the boundary data are sufficiently small, then there exists a unique smooth stationary solution... via the integral equation ϕ=ϕ_b + (1/κ)∫G(r,s)N(ϕ(s))s^{n-1}ds and contraction in X_σ
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear
Vanishing capillarity limit: ∥ρ̃_κ−ρ_+∥_{L^2_r}≤Cκ^{3/4} obtained from energy identity after multiplying by ϕ_κ and integration by parts

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

[1]

Charve and B

F. Charve and B. Haspot,Existence of a global strong solution and vanishing capillarity-viscosity limit in one dimension for the Korteweg system, SIAM J. Math. Anal.,45(2013), 469-494

work page 2013
[2]

Chen,Asymptotic stability of strong rarefaction waves for the compressible fluid models of Korteweg type, J

Z. Chen,Asymptotic stability of strong rarefaction waves for the compressible fluid models of Korteweg type, J. Math. Anal. Appl.,394(2012), 438–448

work page 2012
[3]

Z. Chen, L. He, and H. Zhao,Nonlinear stability of traveling wave solutions for the compressible fluid models of Korteweg type, J. Math. Anal. Appl.,422(2015), 1213–1234

work page 2015
[4]

Danchin and B

R. Danchin and B. Desjardins,Existence of solutions for compressible fluid models of Korteweg type, Ann. Inst. Henri Poincar´ e,18(2001), 97–133

work page 2001
[5]

J. E. Dunn and J. Serrin,On the thermomechanics of interstitial working, Arch. Rational Mech. Anal.,88(1985), 95–133

work page 1985
[6]

Eun, M.-J

N. Eun, M.-J. Kang, and J. Kim,Contraction of viscous-dispersive shocks: Zero viscosity-capillarity limits, arXiv preprint

work page
[7]

Follino, C

R. Follino, C. Lattanzio, and R. G. Plaza,Existence and spectral stability analysis of viscous-dispersive shock profiles for isentropic compressible fluid of Korteweg type, Stud. Appl. Math.,156(2026), e70210

work page 2026
[8]

Follino, R

R. Follino, R. G. Plaza, and D. Zhelyazov,Spectral stability of weak dispersive shock profiles for quantum hydro- dynamics with nonlinear viscosity, J. Differential Equations,359(2023), 330–364

work page 2023
[9]

Germain and P

P. Germain and P. LeFloch,Finite energy method for compressible fluids: the Navier–Stokes–Korteweg model, Commun. Pure Appl. Math.,69(2016), 3–61

work page 2016
[10]

Haspot,Existence of global weak solution for compressible fluid models of Korteweg type, J

B. Haspot,Existence of global weak solution for compressible fluid models of Korteweg type, J. Math. Fluid Mech., 13(2011), 223–249

work page 2011
[11]

Haspot,Global strong solution for Korteweg system with quantum pressure in dimensionN≥2, Math

B. Haspot,Global strong solution for Korteweg system with quantum pressure in dimensionN≥2, Math. Ann., 367(2017), 667–700. 22 KIM

work page 2017
[12]

Huang, I

Y. Huang, I. Hashimoto, and S. Nishibata,On the stability of the spherically symmetric solution to an inflow problem for an isentropic model of compressible viscous flow, arXiv preprint arXiv:2404.07469

work page arXiv
[13]

Hattori and D

H. Hattori and D. Li,Global solutions of a high dimensional system for Korteweg materials, J. Math. Anal. Appl.,198(1996), 84–97

work page 1996
[14]

Han and J

S. Han and J. Kim,Time-asymptotic stability of composite wave for the one-dimensional compressible fluid of Korteweg type, SIAM J. Math. Anal.,58(2026) 547–597

work page 2026
[15]

Han, M.-J

S. Han, M.-J. Kang, J. Kim, and H. Lee,Long-time behavior towards viscous-dispersive shock for Navier–Stokes equations of Korteweg type, J. Differential Equations,426(2025), 317–387

work page 2025
[16]

Hashimoto and A

I. Hashimoto and A. Matsumura,Existence of radially symmetric stationary solutions for the compressible Navier–Stokes equation, Methods App. Anal.28(2021), 299–311

work page 2021
[17]

Hashimoto and A

I. Hashimoto and A. Matsumura,Inviscid limit problem of radially symmetric stationary solutions for compress- ible Navier–Stokes equation, Commun. Math. Phys.,405(2024), article number 220

work page 2024
[18]

Hashimoto and A

I. Hashimoto and A. Matsumura,Existence of radially symmetric stationary solutions for viscous and heat- conductive ideal gas, arXiv preprint arXiv:2507.04234

work page arXiv
[19]

Huang and S

Y. Huang and S. Nishibata,Large-time behaviour of the spherically symmetric solution to an outflow problem for isentropic model of compressible viscous fluid, arXiv preprint arXiv:2308.10206

work page arXiv
[20]

Hashimoto, S

I. Hashimoto, S. Nishibata, and S. Sugizaki,Asymptotic behavior of spherically symmetric solutions to the compressible Navier–Stokes equations towards stationary waves, J. Math. Fluid Mech.26(2024), article number 54

work page 2024
[21]

Jiang,Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain, Commun

S. Jiang,Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain, Commun. Math. Phys.,178(1996), 339–374

work page 1996
[22]

J¨ ungel,Global weak solutions to compressible Navier–Stokes equations for quantum fluids, SIAM J

A. J¨ ungel,Global weak solutions to compressible Navier–Stokes equations for quantum fluids, SIAM J. Math. Anal.,42(2010), 1025–1045

work page 2010
[23]

D. J. Korteweg, Sur la forme que prennent les ´ equations du mouvements des fluides si l’on tient compte des forces capillaires caus´ ees par des variations de densit´ e consid´ erables mais connues et sur la th´ eorie de la capillarit´ e dans l’hypoth` ese d’une variation continue de la densit´ e, Arch. Neerl. Sci. Exact. Nat.,6(1901), 1–24

work page 1901
[24]

Lacroix-Violet and A

I. Lacroix-Violet and A. Vasseur,Global weak solutions to the compressible quantum Navier–Stokes equation and its semi-classical limit, J. Math. Pures Appl.,114(2018), 191–210

work page 2018
[25]

Nakamura and S

T. Nakamura and S. Nishibata,Large-time behavior of spherically symmetric flow of heat conductive gas in a field of potential forces, Indiana Univ. Math. J.,57(2008) 1019–1054

work page 2008
[26]

Nakamura, S

T. Nakamura, S. Nishibata, and S. Yanagi,Large-time behavior of spherically symmetric solutions to an isentropic model of compressible viscous fluid in a field of potential forces, Math. Models Methods App. Sci.,14(2004) 1849–1879

work page 2004
[27]

J. D. van der Waals,Thermodynamische theorie der kapillarit¨ at unter voraussetzung stetiger diche¨ anderung, Z. Phys. Chem.,13(1894), 657–725. (Jeongho Kim) Department of Applied Mathematics, Kyung Hee University, 1732, Deogyeong-daero, Giheung-gu, Yongin-si, Gyeonggi-do 17104, Republic of Korea Email address:jeonghokim@khu.ac.kr

work page