An Eulerian-Lagrangian Formulation of the Compressible Euler Equations with Vacuum

Christian Olivera; Wladimir Neves

arxiv: 2409.13414 · v4 · pith:55PZPNXVnew · submitted 2024-09-20 · 🧮 math.AP

An Eulerian-Lagrangian Formulation of the Compressible Euler Equations with Vacuum

Wladimir Neves , Christian Olivera This is my paper

Pith reviewed 2026-05-23 21:06 UTC · model grok-4.3

classification 🧮 math.AP

keywords compressible Euler equationsvacuumLagrangian flow mapshort-time existenceisentropic flowcompactly supported datageneral pressure lawEulerian-Lagrangian formulation

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The pith

A Lagrangian flow map formulation yields short-time solutions to the compressible Euler equations with vacuum for compactly supported data.

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

The paper develops an Eulerian-Lagrangian formulation of the compressible isentropic Euler equations that incorporates the Lagrangian flow map. This approach establishes the existence of short-time solutions when the pressure law is general and the initial density has compact support, which creates vacuum regions. The method works without any special symmetrization of the equations. Readers would care because vacuum boundaries have long posed technical obstacles in compressible fluid models, and prior techniques often required extra structure that this formulation avoids.

Core claim

The paper presents a novel Eulerian-Lagrangian formulation for the compressible isentropic Euler equations with vacuum. Using the developed Lagrangian flow map formulation, it shows a short-time solution for a general pressure law. The formulation is well defined in the presence of vacuum, namely for compactly supported initial data which constitute an important problem in gas dynamics, and it does so without relying on any special symmetrization.

What carries the argument

The Lagrangian flow map formulation, which tracks particle paths while retaining an Eulerian description of the velocity and density fields to handle vacuum regions.

If this is right

Short-time existence holds for any general pressure law without additional structural assumptions.
The result covers the physically relevant case of compactly supported initial densities that generate vacuum.
The same formulation applies directly to the isentropic system without symmetrization steps.
The approach is defined on the whole space even when the density vanishes outside a bounded set.

Where Pith is reading between the lines

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

The method might adapt to systems with more general equations of state or to related models such as the Euler-Poisson system.
Numerical schemes that discretize along approximate flow maps could inherit stability properties from the continuous formulation.
If the short-time regularity can be continued, the result might inform criteria for global existence or singularity formation.
The formulation could serve as a bridge between purely Eulerian energy methods and purely Lagrangian particle methods.

Load-bearing premise

The Lagrangian flow map stays regular enough and invertible for a short time interval when the initial density has compact support.

What would settle it

An explicit example of compactly supported initial data and a general pressure law where the flow map ceases to be invertible or loses the required regularity in arbitrarily short time would falsify the short-time existence result.

read the original abstract

In this paper, we present a novel Eulerian-Lagrangian formulation for the compressible isentropic Euler equations with vaccum. Using the developed Lagrangian flow map formulation, we show a short-time solution for a general pressure law. A particularly appealing feature of the approach used, it is well defined in the presence of vacuum, namely for compactly supported initial data which constitute an important problem in gas dynamics. Moreover, it does so without relying on any special symmetrization. While analogous results are well understood for incompressible fluids, the compressible setting, particularly in the presence of vacuum, remained open.

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 new Eulerian-Lagrangian formulation that yields short-time existence for compressible Euler with vacuum and general pressure, without symmetrization, but the flow map regularity at the vacuum boundary is the step that needs checking.

read the letter

The paper introduces an Eulerian-Lagrangian formulation of the compressible isentropic Euler equations that is designed to work with vacuum, specifically for compactly supported initial data and a general pressure law. They use this to establish a short-time existence result without relying on symmetrization. This addresses an aspect that was open in the compressible setting, unlike the incompressible case. What the paper does well is to frame the formulation clearly and point out the advantage in handling the vacuum case directly through the flow map. It keeps the approach general by not imposing special structure on the equations. The soft spots center on the regularity of the Lagrangian flow map. When density vanishes, the equation for velocity is just transport, and for completely general p it may be hard to get the necessary bounds on the velocity gradient to ensure the map stays invertible. The abstract does not list specific conditions on the pressure beyond being general, so whether the fixed point argument closes without additional assumptions on p near zero is the part to examine carefully in the proofs. This kind of result is mainly for people in mathematical fluid dynamics who study existence theory for hyperbolic systems with free boundaries or vacuum states. Someone working on similar problems would get value from seeing how the Lagrangian perspective helps here. I recommend sending it for peer review. The claim is interesting enough and the method novel enough that referees should check the details on the map estimates.

Referee Report

1 major / 1 minor

Summary. The paper introduces a novel Eulerian-Lagrangian formulation of the compressible isentropic Euler equations that remains well-defined for compactly supported initial data (vacuum states). It uses this to establish short-time existence of solutions for a general pressure law, without relying on special symmetrization techniques. The approach is presented as extending known incompressible results to the compressible vacuum setting.

Significance. If the central existence result holds, the formulation would address a longstanding open issue in compressible gas dynamics by providing a framework that handles vacuum without artificial symmetrization or restrictive pressure assumptions. This could enable new analyses of free-boundary problems with general equations of state.

major comments (1)

[main existence theorem / flow-map regularity argument] The short-time existence claim for general p (abstract and main theorem): the argument that the Lagrangian flow map X(t,·) remains a C^1 diffeomorphism on short time intervals for compactly supported rho_0 requires velocity-gradient estimates at the vacuum boundary. For arbitrary p without a lower bound on p'(rho) near rho=0, these estimates may fail to close in the fixed-point argument, as the momentum equation reduces to pure transport where rho=0. This is the load-bearing step and must be checked explicitly against the stated generality.

minor comments (1)

[Abstract] Abstract: 'vaccuum' is a typo (should be 'vacuum'). The sentence 'A particularly appealing feature of the approach used, it is well defined...' is grammatically incomplete and should be rephrased for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the central technical point in the short-time existence argument. We address the major comment below.

read point-by-point responses

Referee: [main existence theorem / flow-map regularity argument] The short-time existence claim for general p (abstract and main theorem): the argument that the Lagrangian flow map X(t,·) remains a C^1 diffeomorphism on short time intervals for compactly supported rho_0 requires velocity-gradient estimates at the vacuum boundary. For arbitrary p without a lower bound on p'(rho) near rho=0, these estimates may fail to close in the fixed-point argument, as the momentum equation reduces to pure transport where rho=0. This is the load-bearing step and must be checked explicitly against the stated generality.

Authors: We agree that the vacuum-boundary estimates on the velocity gradient constitute the load-bearing step and thank the referee for requiring an explicit verification against the claimed generality of p. In the fixed-point construction the velocity satisfies the full momentum equation inside the support of rho and reduces to pure transport outside; the C^1 control on the flow map X is obtained from the standard ODE estimate ||∇v(t)||_∞ ≤ ||∇v_0||_∞ exp(∫||∇v||_∞ ds), which is independent of the pressure term and closes for short time solely from the initial C^1 datum and the a-priori bound on the iteration. Because p enters the equation only where rho>0 and p(0)=0 is assumed, no lower bound on p'(rho) near zero is required for the vacuum-region estimate. We will insert a dedicated paragraph (new Remark 3.4) that isolates this vacuum-transport estimate and confirms it holds for any C^2 pressure law with p(0)=0. revision: yes

Circularity Check

0 steps flagged

No circularity: existence result derived from new Eulerian-Lagrangian formulation without self-referential reduction.

full rationale

The paper presents a novel formulation for the compressible Euler equations and proves short-time existence for general pressure laws with vacuum (compactly supported data). No steps reduce by construction to fitted inputs, self-citations, or renamed known results. The central claim is an existence theorem obtained via the flow map, which is independent of the target result. Self-citations, if present, are not load-bearing for the derivation. This matches the default non-circular outcome for a self-contained mathematical existence proof.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only; the result rests on standard PDE regularity assumptions for flow maps and pressure laws that are not detailed here.

axioms (2)

domain assumption Initial data are sufficiently regular and compactly supported to allow a well-defined Lagrangian flow map for short time
Invoked to guarantee the formulation works with vacuum.
standard math Standard local existence theory for ODEs or transport equations applies to the flow map
Background result needed to close the short-time argument.

pith-pipeline@v0.9.0 · 5620 in / 1238 out tokens · 33680 ms · 2026-05-23T21:06:09.633531+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Bahouri, J.Y

H. Bahouri, J.Y. Chemin, , R.Danchin, Fourier Analysis and Nonlin- ear Partial Differential Equations, Grundlehren der mathematischen Wis- senschaften, 2011

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