Gradient blowup of smooth vacuum solutions to 1D compressible Euler equations

Jiaqi Liu; Juhi Jang; Nader Masmoudi

arxiv: 2605.00407 · v1 · submitted 2026-05-01 · 🧮 math.AP

Gradient blowup of smooth vacuum solutions to 1D compressible Euler equations

Juhi Jang , Jiaqi Liu , Nader Masmoudi This is my paper

Pith reviewed 2026-05-09 19:36 UTC · model grok-4.3

classification 🧮 math.AP

keywords compressible Euler equationsvacuum boundarygradient blowupself-similar solutionsfinite time singularityisentropic flowhalf-line domainfree boundary

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The pith

Initially smooth solutions to the isentropic compressible Euler equations on the half-line develop a gradient blowup at the vacuum boundary in finite time.

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

The paper constructs a broad family of solutions to the isentropic compressible Euler equations in one dimension on the half-line. These solutions begin smooth and square-integrable but lose regularity near the vacuum boundary after a finite waiting time, becoming only C^{1-μ} for μ between 1/2 and 1. This produces a blowup in the velocity gradient at the boundary. A sympathetic reader would care because the result shows that smooth initial data for fluid equations with a free vacuum boundary can form singularities in finite time.

Core claim

We construct a large class of solutions that are initially smooth and square-integrable, and which, in finite time, transition to C^{1-μ} regularity for μ ∈ [1/2,1) near the boundary, leading to the gradient blowup at the boundary. It is based on stability analysis of self-similar waiting time solutions recently constructed by the authors.

What carries the argument

Stability of self-similar waiting-time solutions under perturbations that produce the target regularity loss.

Load-bearing premise

The stability analysis of the self-similar waiting-time solutions continues to hold for the class of perturbations used to generate the new solutions.

What would settle it

A numerical computation of the Euler equations with the perturbed initial data that shows the solutions remain smooth and the gradient stays bounded past the predicted time would falsify the claim.

read the original abstract

We consider the isentropic compressible Euler equations in the half-line which govern the motion of gaseous fluids in contact with stationary vacuum boundary. We construct a large class of solutions that are initially smooth and square-integrable, and which, in finite time, transition to $C^{1-\mu}$ regularity for $\mu \in [1/2,1)$ near the boundary, leading to the gradient blowup at the boundary. It is based on stability analysis of self-similar waiting time solutions \cite{JLN2025} recently constructed by the authors.

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 builds a family of smooth initial data for 1D isentropic Euler that develop finite-time gradient blowup at the vacuum boundary by perturbing the authors' earlier self-similar solutions.

read the letter

I read the paper on gradient blowup for smooth vacuum solutions to the 1D compressible Euler equations. The key result is that there exists a large class of initially smooth and square-integrable solutions that lose regularity near the boundary, dropping to C^{1-μ} for μ between 1/2 and 1, which causes the gradient to blow up in finite time. This is done through stability analysis around the self-similar waiting time solutions from the authors' recent prior paper. The new part is the stability-based construction that generates many such solutions rather than just a few. It builds directly on their previous work but adds the perturbation analysis to show how smooth data can evolve into blowup. This is a solid step forward for the field because it gives explicit examples where the solution starts smooth but the vacuum boundary causes gradient blowup, which has been an open issue in the regularity theory. On the positive side, the abstract indicates they have a rigorous way to do this via stability, which if the estimates hold, provides good evidence for the claim. It avoids inventing new techniques and instead leverages the self-similar solutions effectively. The potential issue is in how the perturbations are handled. The stability must apply to perturbations that are tuned to produce the specific regularity loss at the boundary. If those perturbations don't stay within the smallness assumptions or fail to satisfy the weighted decay rates used in the prior work, the argument could have a hole. The stress test points out that the boundary behavior is delicate in those norms, so the paper needs to verify that the chosen perturbations don't violate the conditions for stability to hold. Without seeing the detailed estimates, it's hard to be sure this is fully closed. This paper is for researchers working on hyperbolic PDEs and fluid dynamics, especially those interested in free boundary problems and singularity formation in Euler equations. A reader who knows the background on self-similar solutions would find it useful for seeing how stability can be used to create blowup examples. I would recommend sending it to peer review. The topic is important, the construction is new in this context, and even with the dependence on prior work, it merits a full review to check the technical details.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs a large class of initially smooth and square-integrable solutions to the 1D isentropic compressible Euler equations on the half-line that develop gradient blowup at the vacuum boundary in finite time. These solutions transition from smooth to C^{1-μ} regularity for μ ∈ [1/2,1) near the boundary. The construction is based on a stability analysis of self-similar waiting-time solutions constructed in the authors' prior work [JLN2025].

Significance. If the stability transfer holds, the result would supply explicit examples of smooth vacuum solutions to the compressible Euler system that form singularities at the free boundary in finite time, advancing the understanding of gradient blowup mechanisms in 1D hyperbolic systems with vacuum. The perturbation approach around self-similar solutions is a constructive strength when the weighted-norm controls are verified.

major comments (2)

[Stability analysis and perturbation construction] The central claim relies on transferring the stability theorem from [JLN2025] to a new class of perturbations chosen to induce the C^{1-μ} transition (μ ≥ 1/2) at the boundary. The manuscript does not verify that these perturbations remain inside the smallness ball or satisfy the same decay rates in the weighted Sobolev spaces of the prior work, especially near the vacuum boundary where the weights are most delicate (see the stability analysis section and the construction of the perturbed initial data).
[Main theorem and § on initial data] The initial smoothness and L² integrability of the new solutions are asserted to hold, but without explicit estimates confirming that the boundary perturbations preserve the required function-space membership up to the waiting time, the finite-time transition to C^{1-μ} regularity cannot be rigorously concluded from the prior stability result.

minor comments (2)

The abstract should explicitly name the isentropic compressible Euler system and the half-line domain for clarity.
[References] Reference [JLN2025] should include the arXiv identifier or publication details for completeness.

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 will revise the paper to incorporate the requested verifications and explicit estimates.

read point-by-point responses

Referee: [Stability analysis and perturbation construction] The central claim relies on transferring the stability theorem from [JLN2025] to a new class of perturbations chosen to induce the C^{1-μ} transition (μ ≥ 1/2) at the boundary. The manuscript does not verify that these perturbations remain inside the smallness ball or satisfy the same decay rates in the weighted Sobolev spaces of the prior work, especially near the vacuum boundary where the weights are most delicate (see the stability analysis section and the construction of the perturbed initial data).

Authors: We agree that the current manuscript lacks an explicit verification that the new perturbations remain inside the smallness ball and satisfy the decay rates of the weighted Sobolev spaces from [JLN2025], particularly near the vacuum boundary. In the revised version we will add a dedicated subsection to the stability analysis that constructs the perturbed initial data explicitly and proves that, for sufficiently small parameters, the perturbations satisfy the required smallness and decay conditions in those norms. This will confirm compatibility with the stability theorem and justify the transfer of the C^{1-μ} transition. revision: yes
Referee: [Main theorem and § on initial data] The initial smoothness and L² integrability of the new solutions are asserted to hold, but without explicit estimates confirming that the boundary perturbations preserve the required function-space membership up to the waiting time, the finite-time transition to C^{1-μ} regularity cannot be rigorously concluded from the prior stability result.

Authors: We acknowledge that explicit estimates confirming preservation of smoothness and L² integrability under the boundary perturbations are missing. We will expand the section on initial data to include these estimates, showing that the perturbations (chosen with appropriate support and decay) preserve the required function-space membership. Because the stability theorem in [JLN2025] maintains these properties up to the waiting time, the finite-time transition to C^{1-μ} regularity then follows rigorously from the prior result. revision: yes

Circularity Check

1 steps flagged

Central construction relies on unverified extension of stability from authors' prior self-similar solutions [JLN2025]

specific steps

self citation load bearing [Abstract]
"It is based on stability analysis of self-similar waiting time solutions [JLN2025] recently constructed by the authors."

The new solutions are generated by perturbing the self-similar waiting-time solutions of [JLN2025]. The gradient blowup while preserving initial smoothness and L2 integrability is asserted to follow from stability of those solutions under the chosen perturbations. Because the perturbations are tailored to drive the exact boundary regularity loss that the prior work's weighted spaces were designed to control, the result is load-bearing on an unverified extension of the self-cited stability theorem rather than an independent derivation.

full rationale

The paper's headline result constructs a large class of initially smooth, square-integrable solutions exhibiting finite-time gradient blowup at the vacuum boundary by perturbing self-similar waiting-time solutions. This construction is explicitly based on stability analysis from the authors' immediately preceding paper [JLN2025]. While the stability analysis in the current work is presented as new, the perturbations are chosen precisely to induce the C^{1-μ} transition (μ ≥ 1/2) near the boundary—the regime where the weighted norms and smallness assumptions of the prior stability theorem are most delicate. The manuscript does not provide an independent verification that the chosen perturbations remain inside the stability ball or preserve the required decay rates, so the blowup claim reduces to an assumption that the prior self-cited result extends to this specific perturbation class without additional justification.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and stability properties of the self-similar waiting-time solutions from the authors' previous paper; no additional free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The self-similar waiting-time solutions constructed in JLN2025 exist and admit a stable manifold under suitable perturbations.
The new solutions are obtained as perturbations of these prior profiles; the abstract provides no independent verification of their properties.

pith-pipeline@v0.9.0 · 5388 in / 1273 out tokens · 50947 ms · 2026-05-09T19:36:51.805085+00:00 · methodology

discussion (0)

Reference graph

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Formation of Unstable Shocks for 2D Isentropic Compressible Euler.Commun

Buckmaster, T., Iyer, S. Formation of Unstable Shocks for 2D Isentropic Compressible Euler.Commun. Math. Phys.389, 197–271 (2022). https://doi.org/10.1007/s00220-021-04271-z

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,Falqui, G., Ortenzi, G.,Pedroni, M., Pitton, G., On the “Vacuum” Dam-Break Problem: Exact Solutions and Their Long Time Asymptotics,SIAM J

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Collot, P

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Guo, Y., Hadˇzi´c, M., Jang, J., Larson-Penston self-similar gravitational collapse.Comm. Math. Phys. 386(2021), 1551–1601 40

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Pure Appl

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Hadˇzi´c, M., Jang, J., Expanding large global solutions of the equations of compressible fluid mechanics. Invent. Math., 1–62 (2018)

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Hadˇzi´c, M., Jang, J., A class of global solutions to the Euler-Poisson system.Commun. Math. Phys., 370, no 2, 475–505 (2019)

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Jang and N

J. Jang and N. Masmoudi, Well and ill-posedness for compressible Euler equations with vacuum, Journal of Mathematical Physics 53 (2012), no.11, 115625

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Pure Appl

Jang, J., Masmoudi, N., Well-posedness of compressible Euler equations in a physical vacuum.Comm. Pure Appl. Math.,68, no. 1, 61–111 (2015)

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Waiting time solutions in gas dynamics,Journal de Math ´ematiques Pures et Appliqu´ees,204(2025), https://doi.org/10.1016/j.matpur.2025.103806

Jang, J., Liu, J., Masmoudi, N. Waiting time solutions in gas dynamics,Journal de Math ´ematiques Pures et Appliqu´ees,204(2025), https://doi.org/10.1016/j.matpur.2025.103806

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