Waiting Time Solutions in gas dynamics

Jiaqi Liu; Juhi Jang; Nader Masmoudi

REVIEW 2 major objections 3 minor 1 cited by

Waiting Time Solutions in gas dynamics

T0 review · 2 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper constructs self-similar solutions in which a vacuum interface stays stationary for finite time, then starts moving with physical-vacuum behavior.

desk verdict A solid, novel construction of waiting-time vacuum boundaries for 1D Euler, but Lemma 4.7's barrier argument has a real gap that needs an algebraic verification. read the letter →

arxiv 2501.07831 v1 pith:CA2VDFGW submitted 2025-01-14 math.AP

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

classification math.AP MSC 35Q3135L6576N1035B40

keywords waitingtimesolutionsvacuumfreeboundarycompressibleEulerequationsself-similarphysicalconditionsoniccurvephaseportraitanalysisadiabaticexponent

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that one-dimensional compressible Euler flows with a vacuum boundary can exhibit a waiting time: the interface remains fixed at x=0 for t<0, then at t=0 begins moving along a power-law curve and immediately satisfies the physical vacuum condition. The authors construct, for every adiabatic exponent $\gamma\in(1,3)$ and every similarity exponent $\mu\in(0,1)$, a continuum family of self-similar solutions realizing this transition. These are the first rigorous waiting-time solutions in gas dynamics, confirming a behavior long expected from formal asymptotic studies. The solutions are smooth away from the vacuum boundary and a sonic curve, Hölder continuous at the singular point, and Lipschitz across the sonic curve.

What carries the argument

The central object is the self-similar ODE system (2.7) in the similarity variables $(U,H)$, obtained from the Euler equations via the scaling (2.5)–(2.6). The proof follows a single trajectory in the $(U,H)$ phase plane connecting five critical points: $C$ (stationary boundary for $t<0$), $A$ (the origin in profile variables), $E$ (the point $x=0$ for $t>0$), $D$ (the first sonic point, a weak discontinuity), and $B$ (the moving vacuum boundary, the second sonic point). The trajectory starts on the special solution $H_{sp}(U)=\frac{(\gamma-1)^2}{4}U^2$ — the self-similar simple wave — which carries it from $C$ through $A$ to $D$. At $D$ the Poincaré–Dulac theorem classifies all integral curves through the node: the special solution is the unique curve with slope $C_1$, and every other curve enters with slope $C_2$, producing the Lipschitz (not $C^1$) crossing. The final leg from $B$ to $D$ is obtained by a barrier argument: the curve with the positive unstable slope at $B$ is trapped between the curve $G=0$ and the special solution, forcing it to reach $D$ and thereby close the trajectory.

What would settle it

Numerically integrate the effective ODE (2.9) for a specific pair such as $\gamma=1.1$, $\mu=0.1$, starting from $B=(1,0)$ with slope $C_2=\gamma(1-\mu)/(1+k_2)$, and test whether the resulting curve crosses either the lower boundary $G=0$ or the upper special solution $H_{sp}$ before reaching $U_D=2/(3-\gamma)$; a crossing before $U_D$ would falsify Lemma 4.7 and Theorem 2.3.

Watch

Extended reading notes

Core claim

Theorem 2.3 asserts that for each $\gamma\in(1,3)$ and each $\mu\in(0,1)$ there exist self-similar $(\rho,u)$ solving the vacuum free-boundary Euler equations with $b(t)=0$ for $t<0$ and $b(t)=y_B t^{1/\mu}$ for $t\ge 0$, with $y_B<0$. Before $t=0$ the solution is a simple wave, the special solution $H_{sp}(U)=\frac{(\gamma-1)^2}{4}U^2$, and the stationary boundary has at least $C^1$ velocity and sound speed up to the boundary. After $t=0$ the same special solution extends through the origin, then crosses the first sonic point $D$ with only Lipschitz regularity (a weak discontinuity), and terminates at a second sonic point $B$ that is the moving vacuum boundary, where the physical vacuum condition holds for every $t>0$. The passage through $D$ is forced by the Poincaré–Dulac classification: only one integral curve has the smooth slope at $D$, all others enter with a different slope, and the construction selects the latter to reach the vacuum boundary.

Load-bearing premise

The existence proof depends on the unproven-in-detail claim that the trajectory from the boundary point remains strictly between two comparison curves for the entire interval of $U$ values up to the sonic point; if that global ordering failed for some $\gamma$ near 1, the connecting curve would not reach its target.

Editorial extensions

If this is right

If the theorem is correct, the well-known breakdown of simple waves in Burgers' equation can be resolved inside the full Euler equations: the self-similar Burgers solution continues after its singularity as a Hölder continuous Euler solution with a moving physical-vacuum boundary.
The construction provides a rigorous example where the physical vacuum condition emerges dynamically from initial data that do not satisfy it, supporting the conjecture that such transitions occur generically for vacuum states.
Because finite speed of propagation and far-field cutoffs apply, the local self-similar solutions can be glued to produce finite-energy solutions exhibiting the same waiting-time transition.
The sonic curve emanating from the singular point is a new type of weak discontinuity in these flows, giving an explicit mechanism for loss of regularity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same phase-portrait mechanism may extend to other similarity exponents $\mu$ outside $(0,1)$ or to radial symmetry, where the analogous sonic-point classification would determine whether waiting-time behavior persists.
A direct numerical check of the barrier inequality in Lemma 4.7 over the full parameter range would either confirm the global ordering or uncover a counterexample near $\gamma$ close to 1; the paper only performs local slope comparisons.
The construction suggests that the waiting time is not a special accident but a robust phenomenon selected by the requirement that the continuation be a weak solution across $t=0$; Lemma 3.7 shows any other extension through $A$ creates a jump that violates the weak formulation.
One could test whether the constructed solutions are stable under small perturbations of the initial profile, a question not addressed by the existence proof.

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Formalized claims in Lean

Claim #1: Theorem 2.3 asserts that for each $\gamma\in(1,3)$ and each $\mu\in(0,1)$ there exist self-similar $(\rho,u)$ solving the vacuum free-boundary Euler equations with $b(t)=0$ for $t<0$ and $b(t)=y_B t^{1/\mu}$ for $t\ge 0$, with $y_B<0$. Before $t=0$ the solution is a simple wave, the special solution $H_{sp}(U)=\frac{(\gamma-1)^2}{4}U^2$, and the stationary boundary has at least $C^1$ velocity and

/-- @claim 1 Theorem 2.3 asserts that for each $\gamma\in(1,3)$ and each $\mu\in(0,1)$ there exist self-similar $(\rho,u)$ solving the vacuum free-boundary Euler equations with $b(t)=0$ for $t<0$ and $b(t)=y_B t^{1/\mu}$ for $t\ge 0$, with $y_B<0$. Before $t=0$ the solution is a simple wave, the special solution $H_{sp}(U)=\frac{(\gamma-1)^2}{4}U^2$, and the stationary boundary has at least $C^1$ velocity and  -/
def central_claim : Prop :=

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing to know: this is a genuinely new result. It gives the first rigorous self-similar waiting-time solutions for compressible Euler with physical vacuum after the boundary starts moving, and it does so for all γ∈(1,3) and μ∈(0,1). The construction is concrete: a Burgers/simple-wave leg C→A→E→D, then a new D→B trajectory that meets the vacuum boundary. The weak discontinuity across the sonic curve is a nice qualitative feature.

What's good. The paper does real work. The special solution reduces to Burgers, Lemma 3.7 uses the weak formulation to argue the continuation through A is unique among self-similar ODE continuations, and the Poincaré–Dulac analysis at D correctly identifies the two branches and shows only the non-special branch can reach B. The expansions near B and D are carefully derived. I checked the main line: it's coherent. I also agree the proof is self-contained modulo standard ODE theory; the free parameters are explicit.

The soft spot is Lemma 4.7, the connection from B to D. The proof puts H(U) between H_G (the G=0 curve) and H_sp and says both are barriers. The upper bound H_G0 along H_G on (1,U_D). That positivity is not proven anywhere; if F were negative there, H'→−∞ and the curve could cross. Likewise H_sp>H_G on the whole interval is not shown. These are algebraic inequalities in γ and μ, likely true, but they are not written out. It's a proof obligation, not a physical assumption, and it is exactly what keeps the unique orbit from leaving the corridor. A referee should ask for this as a required revision.

Minor: there's a sign slip in the display of ds/dy in Lemma 4.2 (plus H instead of minus), but the subsequent M computation uses the correct Δ derivative.

Bottom line: the central claim is probably right, and the structure is publishable. The paper deserves a serious referee; the gap in 4.7 is repairable with an explicit algebraic lemma. I'd send it to review but ask for the barrier verification before acceptance.

Referee Report

2 major / 3 minor

Summary. The paper constructs a two-parameter family of self-similar waiting-time solutions for the one-dimensional isentropic compressible Euler equations with vacuum, for adiabatic exponents γ∈(1,3) and similarity exponents μ∈(0,1). In the constructed solutions the vacuum boundary is stationary for t<0, the velocity and sound speed have at least C^1 regularity up to the boundary, and at t=0 the boundary begins to move according to b(t)=y_B t^{1/μ} while satisfying the physical vacuum condition for every t>0. A weak discontinuity emerges from the singular point along the sonic curve. The proof reduces the PDE to the self-similar ODE system (2.7), follows the explicit parabolic special solution H_sp from C to A to E to D, and then uses a phase-portrait analysis and a barrier argument to construct the unique orbit from B to D. Theorem 2.3 summarizes the existence and regularity statements.

Significance. If the proof gaps identified below are repaired, this is the first rigorous construction of waiting-time solutions in gas dynamics in which a stationary vacuum interface changes after finite time into a moving interface with the physical vacuum condition. The paper has substantial technical content: the explicit Burgers special solution is used in a clean way, Lemma 3.7 gives a genuine weak-solution uniqueness statement for the continuation through the point A, and the Poincaré-Dulac analysis at D is detailed and coherent. The construction also yields a genuinely two-parameter family of solutions, with μ and an integration constant as free parameters. The main weakness is localized in Lemma 4.7: the global barrier ordering that is the crux of the B-to-D connection is asserted rather than proved. This is a proof obligation, not a physical assumption, and it appears fixable by direct algebra.

major comments (2)

[Lemma 4.7] The proof of the B–D connection is incomplete at the crucial step where a first intersection with the lower barrier H_G is ruled out. The text states that at a first intersection U_r one has lim_{U→U_r} H'(U)=+∞, but this limit is +∞ only if F(H_G(U),U)>0 on (1,U_D). Since G is positive just above G=0, the sign of H'=F/G is the sign of F; if F were negative at the intersection, one would have H'→−∞ and a transversal crossing of G=0 would not be excluded. The needed positivity of F on the curve G=0 is a finite algebraic inequality in (γ,μ) and is not supplied. The upper-barrier claim H<H_sp is likewise asserted without proof; it can be justified from H(1)<H_sp(1), uniqueness of solutions of (2.9), and the fact that the two curves meet only at D, but none of this appears. Finally, the conclusion H(U_D)=H_D presupposes that the solution exists up to U_D; this can be obtained from the two barriers together with the observation that H_G(U)>(U−1)^2 on (1,U_D), but that observation is also absent. These omissions are load-bearing because Lemma 4.7 is the only argument that the unique curve from B reaches D.
[Equations (3.15), (3.23), (3.25), (3.35), (3.36)] As printed, the displayed formulas for the special solution have the exponent of |U_sp| with the opposite sign from the one obtained by integrating (3.21). Integration of dz/dU=−1/(μU)+(1/μ−1)/(U−U_C) gives |y|=K|U_sp|^{-1/μ}|U_sp−U_C|^{1/μ−1}, not K|U_sp|^{1/μ}|U_sp−U_C|^{1/μ−1}. With the printed exponent, the limits as y→0 and y→∞ in Lemmas 3.6 and 3.9 are interchanged, so those lemmas do not follow from the formulas as stated. The asymptotic formulas (3.26)–(3.27) and the calculation in Section 5 are consistent with the corrected exponent, so this appears to be a typographical error, but it should be fixed throughout.

minor comments (3)

[Section 4, Eq. (4.5)] The denominator in the formula for ds/dy is written as y((U−1)^2+H); to match the expansion of Δ=(U−1)^2−H and the subsequent singularity it should presumably be y((U−1)^2−H).
[Lemma 4.6] The displayed derivative of H_G(U) has a typo: the second term in the numerator should be k_2(3U^2−2(1+μ)U+μ), not k_2(3U−2(1+μ)U+μ k_2). With the corrected formula the claimed positivity for U≥1 is straightforward to verify.
[Section 5, Proof of Theorem 2.3] The sentence 'Property 3 is a consequence of Property 2 and Lemma 4.5' is imprecise; the Hölder continuity up to O rests on the behavior at A and E as well, and the reader would benefit from an explicit reference to the expansions in Section 3 and Lemma 4.8.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the waiting-time construction is derived from an explicit self-similar ODE analysis with external ODE theory, not from fitted inputs or load-bearing self-citations.

full rationale

The paper's derivation chain is self-contained. Theorem 2.3 is obtained by constructing a trajectory of the self-similar ODE system (2.7) through the critical points C, A, E, D, B. The special solution H_sp(U)=((γ-1)^2/4)U^2 is verified by direct substitution (Lemma 2.2), and the continuations C->A, A->E, and E->D are given by explicit formulas (3.15), (3.25), (3.36). The uniqueness of the extension through A is derived from the weak formulation of the Euler equations in Lemma 3.7, not assumed or imported from the authors' earlier works. The key B->D connection rests on the barrier argument in Lemma 4.7, which compares the sought curve with the upper barrier H_sp and the lower barrier H_G. The global verification of the ordering H_G<H<H_sp is compact and would benefit from an explicit algebraic check of F>0 on H_G, but this is a proof-obligation gap or rigor concern, not circularity: the curve is not defined in terms of itself, and no fitted parameter is renamed as a prediction. The classification of integral curves through D uses the Poincare-Dulac theorem cited to the external reference [1] (Arnold), and the saddle classification at B is standard local ODE theory. Self-citations to the authors' prior work appear only as motivation or background (e.g., [18], [19], [20]) and do not carry the construction. The parameters μ and the integration constant K are genuine degrees of freedom that parameterize the continuum family, while the boundary law b(t)=y_B t^{1/μ} emerges from the constructed trajectory rather than being imposed to match a target. There are no fitted data, no self-definitional identifications, and no prediction that reduces to an input by construction. Hence the circularity score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The construction rests on standard ODE theory and the physical model; the only undetermined constants are the family parameters mu and K, which are explicit degrees of freedom rather than fitted values. No new physical entities are introduced.

free parameters (2)

mu (self-similar exponent, mu=1/delta) = arbitrary in (0,1)
Parameter of the family; not fitted to data. It controls boundary geometry and initial Holder regularity (Remark 2.4 and (3.22)).
K (integration constant, equivalently c1) = arbitrary positive
Integration constant in (3.15) and (3.25); sets the coefficient of the next-order term in the initial data (3.22), giving the continuum for fixed mu. This is a degree of freedom, not a fit.

assumptions (5)

standard math Poincare-Dulac theorem and local normal form for ODE systems near equilibrium
Used in Lemma 4.2 to classify all integral curves near D and to claim the only slope-C1 curve is H_sp; imported from Arnold without proof.
standard math Classical local stable and unstable manifold theory for saddles
Used in Lemma 4.5 to assert exactly one integral curve through B in each direction.
standard math Uniqueness and smooth dependence of solutions to the autonomous ODE (2.9) away from singularities
Used in several places to rule out crossings, for example in Lemma 4.1 branch selection and Lemma 4.7 upper barrier argument.
standard math Weak solution definition of Euler equations via integration against test functions
Lemma 3.7 uses integration by parts and continuity of conserved quantities across t=0 to force the special extension; this is the standard weak formulation of (1.1).
domain assumption Polytropic gas law and physical vacuum boundary condition are the admissible model
The paper works within the gamma-law Euler system, the entropy normalization (1.3), and the physical vacuum condition (1.8); these define the problem being solved.

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Pith. "Pith review of Waiting Time Solutions in gas dynamics." pith.science (2026). https://pith.science/paper/CA2VDFGW

@misc{pith2026250107831,
  author       = {Pith},
  title        = {Pith review of: Waiting Time Solutions in gas dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CA2VDFGW}},
  note         = {Machine review of arXiv:2501.07831}
}

abstract

In this article, we construct a continuum family of self-similar waiting time solutions for the one-dimensional compressible Euler equations for the adiabatic exponent $\ga\in(1,3)$ in the half-line with the vacuum boundary. The solutions are confined by a stationary vacuum interface for a finite time with at least $C^1$ regularity of the velocity and the sound speed up to the boundary. Subsequently, the solutions undergo the change of the behavior, becoming only H\"{o}lder continuous near the singular point, and simultaneously transition to the solutions to the vacuum moving boundary Euler equations satisfying the physical vacuum condition. When the boundary starts moving, a weak discontinuity emanating from the singular point along the sonic curve emerges. The solutions are locally smooth in the interior region away from the vacuum boundary and the sonic curve.

Figures

Figures reproduced from arXiv: 2501.07831 by the authors.

**Figure 2.** γ = 1.816, µ = 0.716, Dashed line: Hsp = (γ−1)2 4 U 2 , Red line: F = 0, Blue line: G = 0, Green line: ∆ = 0. D = (UD, HD) = ( 2 3 − γ ,( γ − 1 3 − γ ) 2 ). (3.2) The triple point B is the saddle for y = yB < 0, and the triple point D is the node for y = yD < 0. Zeros of F = G = 0 and ∆ ̸= 0: A = (UA, HA) = (0, 0), (3.3) C = (UC , HC ) = ( 2 γ + 1 µ,( γ − 1 γ + 1 ) 2 µ 2 ), (3.4) E = (UE, HE) = (∞, ∞). (3.5) The poi… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Physical Vacuum Problems for the Full Compressible Euler Equations: Low-regularity Hadamard-style Local Well-posedness
math.AP 2024-11 accept novelty 7.0 of 10

A low-regularity Hadamard-style local well-posedness theorem is proved for the full compressible Euler equations with a physical vacuum boundary in all space dimensions.

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Works this paper leans on

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Arnold V I, Geometrical methods in the theory of ordinary differential equations, Springer Science & Business Media, 2012

work page 2012

[2] [2]

Caffarelli, A

L. Caffarelli, A. Friedman, Regularity of the free boundary for the one-dimensional flow of gas in a porous medium. Amer. J. Math. 101 (1979), no. 6, 1193–1218

work page 1979

[3] [3]

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

Camassa, R. ,Falqui, G., Ortenzi, G.,Pedroni, M., Pitton, G., On the “Vacuum” Dam-Break Problem: Exact Solutions and Their Long Time Asymptotics, SIAM J. Appl. Math. 80 (2020), no. 1, 44–70, DOI 10.1137/19M1266836. MR4046787

doi:10.1137/19m1266836 2020

[4] [4]

Coutand, D., Shkoller, S., Well-posedness in smooth function spaces for the moving boundary three- dimensional compressible Euler equations in physical vacuum. Arch. Ration. Mech. Anal., 206, no. 2, 515–616 (2012)

work page 2012

[5] [5]

Collot, T.-E

C. Collot, T.-E. Ghoul, N. Masmoudi, Singularity formation for Burgers equation with transversal viscosity ,79 Ann. Sci. Ec. Norm. Sup´er, 2022

work page 2022

[6] [6]

Eggers, J., Fontelos, M. A. (2008). The role of self-similarity in singularities of partial differential equations. Nonlinearity. 22(1), R1

work page 2008

[7] [7]

Grundy and V.A

R.E. Grundy and V.A. Bell, Waiting time solutions of the shallow water equations, Proc. R. Soc. Lond. A (1993) 441, 641-648

1993

[8] [8]

der zylinderachse, Luftfahrtforschung 19 (1942), 302–311

Guderley , G., Starke kugelige und zylindrische verdichtungsst¨osse in der n¨ahe des kugelmittelpunktes bzw. der zylinderachse, Luftfahrtforschung 19 (1942), 302–311

work page 1942