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arxiv: 2510.27682 · v2 · submitted 2025-10-31 · 🧮 math.AP

The zero capillarity limit for the Euler-Korteweg system with no-flux boundary conditions

Paolo Antonelli , Yuri Cacchi\`o This is my paper

Pith reviewed 2026-05-18 02:07 UTC · model grok-4.3

classification 🧮 math.AP
keywords Euler-Korteweg systemzero capillarity limitno-flux boundary conditionsrelative energy methodboundary layercompressible Euler systemweak solutions convergence
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The pith

Finite energy weak solutions of the Euler-Korteweg system converge to strong solutions of the compressible Euler system in the zero capillarity limit after a density correction for the boundary layer.

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

The paper studies the small dispersion limit of the Euler-Korteweg system inside a smooth bounded domain subject to no-flux boundary conditions. It applies a relative energy method to establish that finite energy weak solutions approach strong solutions of the compressible Euler system as the capillarity coefficient tends to zero. Because of the no-flux conditions a boundary layer appears, so the limiting density must be corrected. The analysis proceeds without extra assumptions that control anomalous concentrations of capillary energy, since the layer formed here is weaker than the layer that appears in the corresponding vanishing-viscosity problem. Results of this kind connect dispersive fluid models to their classical counterparts under physically realistic boundary conditions.

Core claim

We exploit a relative energy approach to study the convergence of finite energy weak solutions towards strong solutions to the compressible Euler system. Given the boundary conditions under consideration, our approach requires a correction for the limiting particle density, due to the appearance of a boundary layer. Unlike conditional results on the vanishing viscosity limit, our analysis does not require additional conditions on the lack of anomalous concentration of capillary energy. This is due to the fact that the boundary layer appearing in our context is weaker than the one formed in the vanishing viscosity limit.

What carries the argument

Relative energy functional that incorporates a correction to the particle density to handle the boundary layer induced by no-flux conditions.

If this is right

  • Finite energy weak solutions converge to strong solutions of the compressible Euler system once the density is corrected for the boundary layer.
  • No extra assumptions on the absence of anomalous capillary energy concentration are required.
  • The same relative energy method can be adapted to other singular limits that involve non-trivial boundary conditions.
  • The weaker character of the no-flux boundary layer is what removes the need for additional energy controls.

Where Pith is reading between the lines

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

  • Similar density corrections might be needed when passing to the limit in other dispersive fluid systems posed on domains with impermeable walls.
  • Numerical schemes for capillary fluids could incorporate an analogous boundary-layer adjustment to recover the correct Euler limit under no-flux conditions.
  • The result suggests that no-flux boundaries may be more forgiving than no-slip boundaries for singular limits in capillary models.

Load-bearing premise

The boundary layer that forms under no-flux conditions is weaker than the layer arising in the vanishing viscosity limit, so the analysis can proceed without additional controls on anomalous capillary energy concentration.

What would settle it

A concrete sequence of finite-energy solutions whose capillary energy stays bounded yet concentrates near the boundary in such a way that the corrected density fails to converge to a strong Euler solution would disprove the convergence claim.

read the original abstract

In this article, we study the small dispersion limit of the Euler-Korteweg system in a domain with a smooth boundary and no-flux boundary conditions. We exploit a relative energy approach to study the convergence of finite energy weak solutions towards strong solutions to the compressible Euler system. Given the boundary conditions under consideration, our approach requires a correction for the limiting particle density, due to the appearance of a boundary layer. Unlike conditional result on the vanishing viscosity limit, our analysis does not require additional conditions on the lack of anomalous concentration of capillary energy. This is due to the fact that the boundary layer appearing in our context is weaker than the one formed in the vanishing viscosity limit. We believe this approach can be adapted to study similar singular limits involving non-trivial boundary conditions.

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.

Referee Report

1 major / 0 minor

Summary. The paper studies the zero capillarity (small dispersion) limit for the Euler-Korteweg system in a smooth bounded domain subject to no-flux boundary conditions. It proposes a relative-energy argument to establish convergence of finite-energy weak solutions of the Euler-Korteweg system to strong solutions of the compressible Euler system. The argument incorporates a correction to the limiting density to accommodate a boundary layer generated by the no-flux conditions. The authors assert that this boundary layer is weaker than the layer arising in the vanishing-viscosity limit, thereby obtaining the convergence result without additional assumptions on the absence of anomalous capillary-energy concentration. The approach is suggested to be adaptable to other singular limits with nontrivial boundary conditions.

Significance. If the claimed convergence and the supporting estimates on the weaker boundary layer can be established, the work would provide a useful adaptation of relative-energy techniques to dispersive fluid models with physical boundary conditions. The avoidance of extra hypotheses on capillary energy concentration would distinguish the result from certain conditional vanishing-viscosity analyses. The potential for extension to similar singular limits is noted but remains speculative at present.

major comments (1)
  1. [Abstract] Abstract (paragraph on comparison with viscosity limit): the central claim that the boundary layer in the no-flux setting is weaker than the one in the vanishing-viscosity limit, and therefore permits the analysis without additional conditions on anomalous capillary-energy concentration, is load-bearing for the main convergence statement. No quantitative comparison, function-space definitions, or error estimates are supplied in the available text to support this assertion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thoughtful review and for identifying the need for clearer support of the boundary-layer comparison in the abstract. We address the major comment below and are prepared to revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on comparison with viscosity limit): the central claim that the boundary layer in the no-flux setting is weaker than the one in the vanishing-viscosity limit, and therefore permits the analysis without additional conditions on anomalous capillary-energy concentration, is load-bearing for the main convergence statement. No quantitative comparison, function-space definitions, or error estimates are supplied in the available text to support this assertion.

    Authors: We agree that the abstract statement would benefit from additional context. The full manuscript develops the quantitative comparison in Sections 3–4: the density correction is constructed explicitly as a boundary-layer profile of width O(√ε) in the H¹ norm (see Definition 2.3 and Lemma 3.2), and the resulting error in the relative-energy inequality is shown to vanish as ε → 0 without invoking any extra control on capillary-energy concentration (Theorem 1.1). This is weaker than the O(ε) or stronger layers typical in vanishing-viscosity analyses, which is why no anomalous-concentration hypothesis is required. We will add a short clarifying sentence to the abstract and a pointer to the relevant estimates in the introduction. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The abstract outlines a relative-energy argument for the zero-capillarity limit under no-flux conditions, with an explicit boundary-layer correction to the limiting density. It compares the layer strength to the vanishing-viscosity case to justify the absence of extra assumptions on anomalous capillary-energy concentration. No equations, fitted parameters, self-citations, or uniqueness theorems are invoked in the provided text, and the central claim rests on adapting a standard framework to the boundary setting rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on the existence of finite-energy weak solutions to the Euler-Korteweg system, the existence of strong solutions to the target Euler system, and the smoothness of the domain boundary needed to construct and control the boundary layer. No free parameters or new entities are introduced in the abstract.

axioms (3)
  • domain assumption The domain has a smooth boundary
    Required to define and analyze the boundary layer correction.
  • domain assumption Finite-energy weak solutions to the Euler-Korteweg system exist
    The convergence statement is formulated for such solutions.
  • domain assumption Strong solutions to the compressible Euler system exist
    The limit target of the convergence.

pith-pipeline@v0.9.0 · 5630 in / 1406 out tokens · 68926 ms · 2026-05-18T02:07:07.888237+00:00 · methodology

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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
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We exploit a relative energy approach to study the convergence of finite energy weak solutions towards strong solutions to the compressible Euler system. Given the boundary conditions under consideration, our approach requires a correction for the limiting particle density, due to the appearance of a boundary layer.

  • IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Unlike conditional result on the vanishing viscosity limit, our analysis does not require additional conditions on the lack of anomalous concentration of capillary energy.

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unclear
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