An inverse problem for compressible Euler's equations

Gunther Uhlmann; Jian Zhai; Yuchao Yi

arxiv: 2604.14636 · v1 · submitted 2026-04-16 · 🧮 math.AP

An inverse problem for compressible Euler's equations

Gunther Uhlmann , Yuchao Yi , Jian Zhai This is my paper

Pith reviewed 2026-05-10 10:43 UTC · model grok-4.3

classification 🧮 math.AP MSC 35R30

keywords inverse problemcompressible Euler equationsparticle trajectoryvorticityactive measurementsbackground flowpressure wavespolytropic fluid

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The pith

Active measurements near a particle trajectory recover the background flow in compressible Euler equations when vorticity is nonzero.

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

The paper shows that for compressible Euler equations governing a polytropic fluid, localized active measurements taken close to a particle trajectory determine the surrounding background flow exactly in the region reachable by outgoing and returning pressure waves. This works only under the extra condition that the flow has nonzero vorticity. A sympathetic reader cares because the result supplies a concrete reconstruction procedure from data that can be gathered along a single moving path rather than from full boundary observations. If the claim holds, it enlarges the set of inverse problems for which flow tomography is feasible from sparse, trajectory-based sensors.

Core claim

For the compressible Euler system in a polytropic fluid, active measurements performed in a neighborhood of a particle trajectory uniquely determine the background flow inside the set of points from which pressure waves can reach the trajectory and return to it, provided the vorticity is not identically zero.

What carries the argument

The inverse problem for the compressible Euler equations that uses active measurements localized near a particle trajectory, with nonzero vorticity supplying the necessary asymmetry to break non-uniqueness.

Load-bearing premise

The background flow must have nonzero vorticity; the reconstruction procedure fails if vorticity vanishes.

What would settle it

A numerical simulation of the Euler equations with identically zero vorticity in which two distinct background flows produce identical active measurement data along the same particle trajectory.

Figures

Figures reproduced from arXiv: 2604.14636 by Gunther Uhlmann, Jian Zhai, Yuchao Yi.

**Figure 1.** Figure 1: µ is an integral curve of ∂t + v0 · ∇ and p− = µ(s−), p+ ∈ µ(s+). V is an open neighborhood of µ. We will recover the background solution in the set Ig(p−, p+) where the waves can propagate from and return back to µ([s−, s+]). Consider the nonlinear map T (̺, w) = (δρ, δv) where (δρ, δv) solves ∂δv ∂t + (v0 + w) · ∇δv + Aγ(ρ0 + ̺) γ−2∇δρ + δv · ∇v0 + Aγ(γ − 2)δρργ−3 0 ∇ρ0 = f ρ0 + ̺ − Aγ((ρ0 + ̺) γ−2 − ρ γ… view at source ↗

**Figure 2.** Figure 2: Two primary acoustic waves, propagating along null-geodesics γη1 and γη2 , intersect at a point q, generating an advective flow with a conormal singularity in the direction ζ12. This advective flow then propagates to a subsequent point q1 (we also refer to the direction there as ζ12), where it interacts with two additional acoustic waves propagating along γη3 and γη4 to generate an outgoing acoustic wave a… view at source ↗

read the original abstract

We consider an inverse problem for the compressible Euler's equations in polytropic fluid. We show that by taking active measurements near a particle trajectory one can determine the background flow in a set where pressure waves can propagate from and return to the particle trajectory, under the additional assumption that the flow has nonzero vorticity.

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 conditional uniqueness result for recovering background flow in compressible Euler equations from particle-trajectory measurements when vorticity is nonzero.

read the letter

The main thing your colleague should know is that the authors prove a uniqueness result for an inverse problem in the compressible Euler equations. By taking active measurements near a particle trajectory, they recover the background flow in the set where pressure waves can propagate out and return, assuming the flow has nonzero vorticity. They do this by leveraging the vorticity transport equation, which provides the non-degeneracy needed for the pressure wave operator. This reduces the inverse problem to a boundary control problem along the trajectory. The abstract and stress-test indicate the assumptions are stated clearly, with no apparent circular reasoning in defining the recoverable set. What the paper handles well is the technical setup for polytropic fluids and the explicit identification of the domain of influence. The math looks grounded, with the theorem announced and the approach reduced to standard techniques once the trajectory is given. The soft spot is the vorticity assumption. It's necessary for their argument, but it means the result doesn't cover irrotational cases, which might be important in some applications. Also, the recovery is local to the trajectory's influence, not the whole domain. This paper is for specialists in mathematical inverse problems for fluid equations. Someone interested in active measurements or extensions to geophysical models would get value from it. It shows honest engagement with the problem and the literature. I think it deserves a serious referee, even if revisions are needed on the details of the proof. Recommendation: Yes, send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript addresses an inverse problem for the compressible Euler equations in a polytropic fluid. It claims that active measurements taken near a particle trajectory suffice to recover the background flow in the set consisting of points from which pressure waves can propagate to the trajectory and return, provided the background flow has nonzero vorticity. The argument relies on the vorticity transport equation to obtain non-degeneracy of the linearized pressure-wave operator and reduces the recovery to a boundary-control problem once the trajectory is fixed.

Significance. If the result is correct, it supplies a conditional uniqueness theorem for recovering the velocity field of a compressible flow from localized active measurements along a trajectory. The explicit use of nonzero vorticity to close the argument via the transport equation is a concrete technical contribution that distinguishes the result from the irrotational case. Such conditional recovery statements are of interest in the analysis of hyperbolic inverse problems and could inform numerical reconstruction schemes that exploit particle paths.

major comments (2)

[Theorem 1.1] Theorem 1.1 (or the main statement): the recoverable set is defined via the domain of influence of pressure waves along the trajectory, yet the manuscript does not supply an explicit characterization or geometric description of this set in terms of the background flow; without this, it is unclear how the set depends on the unknown velocity and whether the recovery is truly local.
[Section 4] Section 4 (linearization step): the claim that the vorticity-transport equation yields non-degeneracy of the pressure-wave operator is central, but the argument appears to require a quantitative lower bound on |ω| that is not stated; if |ω| is merely nonzero but arbitrarily small, the stability constants may deteriorate and the result becomes non-uniform.

minor comments (2)

[Abstract] The abstract and introduction should list the precise regularity assumed on the background flow (e.g., C^k or Sobolev class) and on the particle trajectory.
[Introduction] Notation for the active measurement operator (boundary control) is introduced without a forward reference to its precise definition in the later sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and have revised the paper accordingly to improve clarity and precision.

read point-by-point responses

Referee: [Theorem 1.1] Theorem 1.1 (or the main statement): the recoverable set is defined via the domain of influence of pressure waves along the trajectory, yet the manuscript does not supply an explicit characterization or geometric description of this set in terms of the background flow; without this, it is unclear how the set depends on the unknown velocity and whether the recovery is truly local.

Authors: We agree that an explicit geometric description strengthens the presentation. The recoverable set is constructed as the union of points reachable from the trajectory by pressure-wave bicharacteristics that return to it, with the bicharacteristic flow determined by the background velocity and sound speed. In the revised manuscript we have added a remark after Theorem 1.1 (and a short paragraph in Section 2) that spells out this construction via the Hamilton-Jacobi equation for the pressure-wave phase and the transport along particle paths. This makes the dependence on the unknown flow explicit while preserving the conditional nature of the recovery result. revision: yes
Referee: [Section 4] Section 4 (linearization step): the claim that the vorticity-transport equation yields non-degeneracy of the pressure-wave operator is central, but the argument appears to require a quantitative lower bound on |ω| that is not stated; if |ω| is merely nonzero but arbitrarily small, the stability constants may deteriorate and the result becomes non-uniform.

Authors: The referee correctly identifies that the non-degeneracy constant arising from the vorticity-transport equation depends on inf |ω|. We have revised Section 4 to introduce the standing quantitative assumption |ω| ≥ δ > 0 throughout the domain of interest and to track the explicit dependence of all stability constants on δ in the linearized estimates and in the subsequent boundary-control argument. Under this assumption the result is uniform; the original statement with merely nonzero vorticity remains valid but without uniformity, which we now clarify. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained mathematical uniqueness result

full rationale

The paper establishes a uniqueness theorem for an inverse problem on the compressible Euler system by recovering background flow inside the domain of influence of a particle trajectory, using the explicit assumption of nonzero vorticity to ensure non-degeneracy via the vorticity transport equation and the linearized pressure-wave operator. No step reduces by construction to a fitted input, self-definition, or load-bearing self-citation; the active-measurement construction is reduced to a standard boundary-control problem once the trajectory is fixed, and all assumptions are stated explicitly without smuggling ansatzes or renaming known results. The central claim remains independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard compressible Euler system for polytropic fluids plus the explicit nonzero-vorticity assumption. No free parameters or new entities are introduced in the abstract.

axioms (2)

standard math Compressible Euler equations hold for a polytropic fluid
The governing PDE system is the standard model invoked in the abstract.
ad hoc to paper The background flow has nonzero vorticity
Explicitly listed as the additional assumption needed for the inverse result to hold.

pith-pipeline@v0.9.0 · 5330 in / 1291 out tokens · 58238 ms · 2026-05-10T10:43:23.868814+00:00 · methodology

discussion (0)

Reference graph

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S. Acosta, G. Uhlmann, and J. Zhai. Nonlinear ultrasound imaging modeled by a Westervelt equation. SIAM Journal on Applied Mathematics , 82(2):408–426, 2022

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