A Two-Phase Free Boundary Problem for Axisymmetric Subsonic Euler Flows with Contact Discontinuities

Hyangdong Park

arxiv: 2605.03714 · v2 · pith:X6RBO4ZTnew · submitted 2026-05-05 · 🧮 math.AP

A Two-Phase Free Boundary Problem for Axisymmetric Subsonic Euler Flows with Contact Discontinuities

Hyangdong Park This is my paper

Pith reviewed 2026-05-07 14:58 UTC · model grok-4.3

classification 🧮 math.AP

keywords contact discontinuitiesaxisymmetric Euler flowssubsonic flowsfree boundary problemsinfinite cylindersHelmholtz decompositionvorticityangular momentum

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The pith

Axisymmetric subsonic Euler flows with vorticity in infinite cylinders admit contact discontinuities.

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

The paper establishes the existence of contact discontinuities for axisymmetric subsonic Euler flows that carry non-zero vorticity and non-zero angular momentum density inside three-dimensional cylinders of infinite length. These discontinuities appear as free surfaces separating two phases of the flow while pressure remains continuous and the flow stays subsonic everywhere. A reader would care because the result shows that steady flows with sharp interfaces can persist indefinitely along a pipe without the interfaces collapsing or the speed becoming sonic. The argument begins by truncating the cylinder to a finite segment, reformulates the equations via Helmholtz decomposition to separate the flow into potential and rotational parts, and solves the resulting two-phase free boundary problem by iteration; the infinite-cylinder solutions are recovered by passing to the limit.

Core claim

We prove the existence of contact discontinuities for axisymmetric subsonic Euler flows with non-zero vorticity and non-zero angular momentum density in three-dimensional infinitely long cylinders. The problem is formulated as a two-phase free boundary problem in a cylinder of infinite length. We first solve the cut-off domain problem and take the limit. The cut-off domain problem is reformulated by using a Helmholtz decomposition method and solved via an iteration method. A sophisticated iteration scheme is devised to solve the two-phase free boundary problem.

What carries the argument

Helmholtz decomposition of the velocity field combined with an iterative scheme that alternately updates the free boundary and the flow variables inside a truncated cylinder, followed by a limiting argument that extends the solution to infinite length.

If this is right

Contact discontinuities can coexist with non-zero vorticity and angular momentum in steady axisymmetric subsonic flows.
The free boundary separating the two phases remains regular and does not introduce new singularities when the cylinder is extended to infinite length.
The Helmholtz decomposition separates the rotational and irrotational parts so that the iteration can enforce the jump conditions and the subsonic constraint simultaneously.
Existence in the infinite cylinder follows directly from uniform estimates on the truncated problems.

Where Pith is reading between the lines

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

The same truncation-and-limit procedure might be adapted to cylinders with slowly varying cross-sections or to annular domains.
These steady solutions could serve as background states for studying the stability of contact discontinuities under small time-dependent perturbations.
If the iteration converges for a wider range of vorticity distributions, the method would yield families of solutions parameterized by the angular momentum density.

Load-bearing premise

The iterative solutions constructed on successively longer finite cylinders converge to a limit that keeps the contact discontinuity intact and the flow subsonic.

What would settle it

A sequence of numerical solutions on cylinders of increasing length in which the size of the velocity or density jump across the interface tends to zero or a sonic point appears inside the flow.

Figures

Figures reproduced from arXiv: 2605.03714 by Hyangdong Park.

**Figure 1.1.** Figure 1.1: Contact discontinuity Γ in a 3-D circular cylinder We are concerned with the existence of contact discontinuities in a three-dimensional circular cylinder Ω of infinite length defined as follows: Ω := (x1, x2, x3) ∈ R 3 : x1 > 0, 0 ≤ q x 2 2 + x 2 3 < 1 . To study flows in a circular cylinder, we use a standard cylindrical coordinate system (x, r, θ) given by the relation x1 = x, x2 = r cos θ, x3 = r… view at source ↗

**Figure 2.1.** Figure 2.1: Left: background state, right: Problem 2.1 The goal of this work is to solve the following problem: Problem 2.1. For given radial functions w ± en : Γ± 0 → R, S ± en : Γ± 0 → R, and v ± en : Γ± 0 → R, find a weak solution U := (ρ ±, u ±, p±) to the Euler system (1.1) in Ω with a contact discontinuity Cf := Ω ∩ {r = f(x)} in the sense of Definition 1.2 such that the following properties hold: (i) ρ ±, u ±… view at source ↗

read the original abstract

We study a free boundary problem for the three-dimensional steady compressible Euler equations in an infinitely long circular cylinder. The free boundary is a contact discontinuity separating two axisymmetric rotational subsonic flows, neither of which is prescribed a priori. The pressure continuity condition couples two unknown Euler states through an unknown interface, leading to a genuinely two-phase free boundary problem. Using a Helmholtz decomposition, we reformulate the pressure continuity condition as nonlinear boundary conditions for the Helmholtz variables. This reformulation reveals a nonlinear coupling among the free boundary, the transport subsystem, and the elliptic subsystem associated with the two Euler phases. To resolve this coupling, we develop a coupled iteration framework in which all components are determined simultaneously. Uniform estimates independent of the truncation length allow us to pass to the infinite-length limit. As a consequence, we prove the existence of contact discontinuities separating two rotational subsonic Euler flows in a three-dimensional infinitely long cylinder.

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 an existence proof for contact discontinuities in axisymmetric subsonic Euler flows with vorticity and angular momentum inside infinite cylinders, but the result stands or falls on whether the iteration yields bounds uniform in the cut-off length.

read the letter

This paper proves existence of contact discontinuities for axisymmetric subsonic Euler flows with non-zero vorticity and angular momentum in three-dimensional infinitely long cylinders. They treat the problem as a two-phase free boundary problem, solve it first on finite cut-off cylinders, and pass to the limit as the cut-off length goes to infinity. The new part is carrying this out while keeping both vorticity and angular momentum non-zero throughout the infinite domain. Most earlier existence results for contact discontinuities stay in bounded regions or drop the rotational terms, so the infinite-cylinder case with those quantities active is a genuine extension. The authors reformulate via Helmholtz decomposition and then run an iteration on the free boundary and the flow variables. That is a standard route for these problems, and the abstract indicates they tuned the iteration to handle the two-phase structure in the cut-off setting. If the finite-domain problems are well-posed under that scheme, the local part of the argument is on solid ground. The soft spot is the passage to the limit. Uniform control on the free-boundary position, the density jump, the velocity, the vorticity, and the angular momentum is required so that the limiting object stays subsonic, keeps the discontinuity, and satisfies the Euler equations globally. Without those cut-off-independent bounds the limit can lose the jump or become sonic at large distances. The paper asserts that the iteration produces such bounds, but that step is load-bearing and must be checked in detail. The rest of the argument does not appear circular; it builds from the decomposition and iteration without fitting quantities to themselves. This is a paper for specialists in mathematical fluid dynamics who work on free-boundary problems for the Euler equations in unbounded domains. A reader who follows existence results for axisymmetric or rotational flows will see a concrete new case. It is not broad enough to interest people outside that niche, but the claim is specific enough that a serious referee can evaluate whether the uniform estimates close. I would send it to peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript proves the existence of contact discontinuities for axisymmetric subsonic Euler flows with non-zero vorticity and non-zero angular momentum density in three-dimensional infinitely long cylinders. The problem is formulated as a two-phase free-boundary problem; the authors first solve a cut-off finite-cylinder version via Helmholtz decomposition and a custom iteration scheme, then pass to the limit as the cut-off length tends to infinity.

Significance. If the uniform estimates hold, the result would constitute a meaningful extension of free-boundary theory for the Euler system to infinite cylindrical domains while accommodating both vorticity and a density jump. The Helmholtz-plus-iteration framework is a standard but technically demanding route for such problems, and a successful implementation would supply a template for related unbounded-domain discontinuity problems.

major comments (1)

[cut-off domain problem and limit process] The central limit argument requires a priori bounds on the free-boundary location, density jump, velocity, vorticity, and angular momentum that remain uniform with respect to the cut-off length. The manuscript must exhibit these bounds explicitly (or derive them from the iteration) and verify that they prevent the limiting flow from becoming sonic or losing the discontinuity; without such uniformity the passage to the infinite cylinder is not justified.

minor comments (2)

Clarify the precise function spaces in which the iteration is performed and state the contraction or convergence criterion used for the scheme.
Add a brief remark on how the axisymmetric assumption is used to control the angular momentum term throughout the iteration.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the importance of uniform estimates in the limiting procedure. We address the major comment below and will revise the manuscript accordingly to make the relevant bounds fully explicit.

read point-by-point responses

Referee: [cut-off domain problem and limit process] The central limit argument requires a priori bounds on the free-boundary location, density jump, velocity, vorticity, and angular momentum that remain uniform with respect to the cut-off length. The manuscript must exhibit these bounds explicitly (or derive them from the iteration) and verify that they prevent the limiting flow from becoming sonic or losing the discontinuity; without such uniformity the passage to the infinite cylinder is not justified.

Authors: We agree that explicit uniform bounds are indispensable for justifying the passage to the infinite cylinder. In our construction, the cut-off problem is solved by a fixed-point iteration on the Helmholtz-decomposed system; the a priori estimates for the free-boundary height function, density jump, velocity, vorticity, and angular momentum are derived from the subsonic condition (Mach number strictly less than 1) and the axisymmetric structure, and these estimates depend only on the far-field data and are therefore independent of the cut-off length L. We will add a dedicated subsection that states these L-independent bounds explicitly and verifies that they keep the limiting flow strictly subsonic with a positive density jump, thereby preserving the contact discontinuity. revision: yes

Circularity Check

0 steps flagged

No circularity: standard cut-off + limit + iteration proof

full rationale

The derivation proceeds by reformulating the two-phase free-boundary problem on a finite cut-off cylinder via Helmholtz decomposition, solving the approximate problem by iteration, and passing to the limit as the cut-off length tends to infinity. This is a conventional existence strategy that does not reduce any claimed result to a definition of itself, a fitted parameter renamed as a prediction, or a self-citation chain. No equations or steps in the provided abstract or description exhibit the enumerated circularity patterns; the central existence claim therefore remains independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof relies on standard mathematical assumptions from PDE theory for the Euler equations and free boundary problems; no free parameters or invented physical entities are introduced in the abstract.

axioms (2)

domain assumption The Euler equations admit a Helmholtz decomposition for axisymmetric flows with the given vorticity and angular momentum.
Invoked to reformulate the two-phase free boundary problem.
domain assumption Solutions to the cut-off domain problem converge to a solution on the infinite cylinder while preserving subsonicity and the contact discontinuity.
Central to taking the limit after solving the truncated problems.

pith-pipeline@v0.9.0 · 5364 in / 1278 out tokens · 80254 ms · 2026-05-07T14:58:39.045342+00:00 · methodology

A Two-Phase Free Boundary Problem for Axisymmetric Subsonic Euler Flows with Contact Discontinuities

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)