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arxiv: 2606.12758 · v1 · pith:SJV4C74Tnew · submitted 2026-06-10 · 🧮 math.AP · math-ph· math.MP· physics.flu-dyn

Self-similar imploding solutions of the 1D compressible Euler equations with a far field cutoff

Jack Luong , Scott Ramsey , Andrea L. Bertozzi , Roy Baty This is my paper

Pith reviewed 2026-06-27 08:42 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MPphysics.flu-dyn
keywords compressible Euler equationsself-similar implosionsrarefaction wavesfar field cutoffKidder solutionGuderley solutionisentropic flowanalytic solutions
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The pith

A constant density cutoff in the far field of Kidder's 1D imploding solution produces an exact analytic form even though a rarefaction suppresses the implosion.

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

The paper studies self-similar imploding solutions of the one-dimensional isentropic compressible Euler equations. Kidder's known closed-form solution is unbounded at large distances, while other smooth imploding solutions are numerically unstable. Replacing the unbounded far field with a constant density cutoff in the initial data produces a non-centered rarefaction wave that damps the implosion. Despite the damping, the modified problem still admits an exact analytic solution, which the authors derive explicitly and confirm by direct numerical simulation.

Core claim

In Kidder's formulation of the radially symmetric isentropic compressible Euler equations in one dimension, the unbounded far-field condition is replaced by a constant density cutoff in the initial data. A non-centered rarefaction then emerges from the cutoff location and suppresses the implosion, yet an exact analytic self-similar solution for the entire flow field can still be constructed and matches numerical results.

What carries the argument

The non-centered rarefaction wave that originates at the far-field density cutoff and suppresses the implosion while preserving an exact self-similar solution.

If this is right

  • The analytic solution remains numerically stable and directly computable.
  • The emerging rarefaction damps the implosion strength at late times.
  • An explicit closed-form expression is available for both density and velocity fields.
  • Numerical simulations reproduce the analytic predictions to high accuracy.

Where Pith is reading between the lines

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

  • The cutoff technique may supply a practical bounded test problem for codes that must handle interacting rarefactions and shocks.
  • Analogous far-field modifications could be tested on other self-similar Euler flows whose unbounded versions are known.
  • The suppression mechanism suggests that controlled far-field data can systematically alter global implosion behavior without destroying analytic tractability.

Load-bearing premise

The initial data with a constant density cutoff produces a non-centered rarefaction that suppresses the implosion while still permitting an exact analytic solution.

What would settle it

Numerical integration of the cutoff initial data that fails to generate the predicted non-centered rarefaction or deviates from the derived analytic profiles would falsify the existence of the exact solution.

Figures

Figures reproduced from arXiv: 2606.12758 by Andrea L. Bertozzi, Jack Luong, Roy Baty, Scott Ramsey.

Figure 1
Figure 1. Figure 1: Normalized density and velocity for various choices of ǫ in initial data (13) at their respective shock times. This paper is organized as follows. We begin by showing how the isentropic, homogeneous compression solution of [19] can be constructed via the principle of the linear velocity ansatz - a mathematical formalization of homogeneous compression - in Section 2. We state the general form of homogeneous… view at source ↗
Figure 2
Figure 2. Figure 2: Density and Velocity past the shock time for ǫ = 1. After the shock occurs, the gas expands outwards into the com￾pressing gas in the far field. velocity solutions, we will deviate from the derivation done in [14] and [31], giving a new derivation of this solution. We introduce the linear velocity ansatz : (14) v(r, t) = r R˙(t) R(t) where R(t) is the scale radius, to be determined. This velocity is a homo… view at source ↗
Figure 3
Figure 3. Figure 3: Predicted characteristics stemming from the cutoff im￾plosion initial data. The characteristics left of the cutoff position x0 (dashed lines) are expected to follow the Lagrangian position described in equation (24). The characteristics right of the cutoff position x0 (solid lines) are expected to remain vertical since the initial velocity and density is constant in this region. A rarefaction is expected t… view at source ↗
Figure 4
Figure 4. Figure 4: Initial data for the cutoff implosion problem. 5. Discussion In this section, we will discuss the results of the numerical simulations shown in Figures 5, 6, and 7 and [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Analytical solution and selected representatives of the computed solution for density at approximate times t = 0.25, 0.5, 0.75 and t = 1. t ρ v 0.25 2.28 · 10−5 2.36 · 10−5 0.50 3.37 · 10−5 3.81 · 10−5 0.75 4.01 · 10−5 5.03 · 10−5 0.99 4.08 · 10−5 6.00 · 10−5 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Analytical solution and selected representatives of the computed solution for velocity as well as the local speed of sound at approximate times t = 0.25, 0.5, 0.75 and t = 1. instills confidence that our proposed analytical solution is correct, coupled with the discussion about unique solutions in the previous section. We next determine various features we expect to see in the numerical simulation. The com… view at source ↗
Figure 7
Figure 7. Figure 7: Computed density and velocity plotted as a function of the similarity variable y = x t+1 at increasing times in the direction of the arrow, as predicted in the analytical expressions for the density and velocity in equations (38) and (39). The approximate times are t = 0, 0.20, 0.33, 0.46, 0.60, 0.73, 0.87 and 0.97. At each time snapshot, the solution collapses onto the same middle region as highlighted in… view at source ↗
Figure 8
Figure 8. Figure 8: Density and Velocity plotted as a function of the similarity variable η = x−2.5 t at increasing times in the di￾rection of the arrow. The approximate times are t = 0, 0.20, 0.33, 0.46, 0.60, 0.73, 0.87 and 0.97. Note the solutions at various time snapshots does not collapse onto the same curve, in￾dicating the solution is not self-similar in η. However, the left and right boundaries of the middle region ar… view at source ↗
read the original abstract

Imploding solutions to the radially symmetric, isentropic, compressible Euler equations have been well-studied, inspired by the work of Guderley. However, these smooth imploding solutions are shown to be numerically unstable and difficult to compute in practice. On the other hand, the imploding solution of Kidder has a closed form solution and is numerically computable. But, it is unbounded in the far field. We consider Kidder's formulation in one dimension in which the unbounded far field condition is replaced with a constant density cutoff of the initial data. Strikingly, a non-centered rarefaction emerges from the cutoff and suppresses the implosion. We present an exact analytic solution to the problem with the cutoff and support our theoretical predictions with numerical simulations.

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.

Referee Report

3 major / 2 minor

Summary. The paper modifies Kidder's closed-form imploding solution of the 1D isentropic compressible Euler equations by replacing the unbounded far-field condition with a constant-density cutoff at finite radius. It asserts that this generates a non-centered rarefaction wave that suppresses the implosion while preserving an exact analytic self-similar form, and supports the claim with numerical simulations.

Significance. If the central claim holds, the result supplies a rare closed-form example of how a far-field cutoff alters implosion dynamics via a simple-wave interaction, which could aid analysis of stability and numerical schemes for the Euler system. The reuse of Kidder's exact base solution and the combination of analytic construction with numerics are strengths.

major comments (3)
  1. [Abstract, §2] Abstract and §2: the assertion of a globally exact analytic self-similar solution after the non-centered rarefaction reaches the core is not accompanied by any derivation showing that the Riemann invariants remain constant across the interaction region while preserving the Guderley/Kidder scaling; the matching condition between the rarefaction fan and the imploding core is not exhibited.
  2. [§3] §3 (construction of the rarefaction): the claim that the cutoff produces a simple wave whose head and tail characteristics remain compatible with a single self-similar variable requires explicit verification that the invariants on the fan identically reproduce the core solution; without this step the solution cannot be guaranteed to stay self-similar once characteristics cross the implosion region.
  3. [Numerical results] Numerical section: the simulations are said to support the analytic prediction, yet no grid resolution, convergence study, or quantitative comparison (e.g., L2 error to the claimed closed-form profile) is reported, leaving open whether the observed suppression is due to the exact analytic mechanism or numerical dissipation.
minor comments (2)
  1. Notation for the cutoff radius and the rarefaction head/tail speeds should be introduced with a single consistent symbol and referenced in both the analytic and numerical sections.
  2. The manuscript would benefit from a brief statement of the isentropic exponent γ used throughout and confirmation that the same value appears in both the analytic construction and the simulations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The points raised correctly identify places where the manuscript would benefit from additional explicit derivations and numerical details. We address each major comment below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [Abstract, §2] Abstract and §2: the assertion of a globally exact analytic self-similar solution after the non-centered rarefaction reaches the core is not accompanied by any derivation showing that the Riemann invariants remain constant across the interaction region while preserving the Guderley/Kidder scaling; the matching condition between the rarefaction fan and the imploding core is not exhibited.

    Authors: We agree that an explicit derivation of the matching conditions and invariance of the Riemann invariants is missing from the current text. In the revised manuscript we will add a short subsection that computes the invariants J± across the rarefaction fan, verifies that one invariant remains constant while the other reproduces the Kidder core values at the interface, and confirms that the head and tail characteristics are compatible with the single self-similar variable ξ = r / t^α. The matching will be shown by direct substitution of the isentropic relations into the characteristic equations. revision: yes

  2. Referee: [§3] §3 (construction of the rarefaction): the claim that the cutoff produces a simple wave whose head and tail characteristics remain compatible with a single self-similar variable requires explicit verification that the invariants on the fan identically reproduce the core solution; without this step the solution cannot be guaranteed to stay self-similar once characteristics cross the implosion region.

    Authors: The construction in §3 proceeds from the observation that the cutoff generates a centered simple wave in the self-similar coordinate. To make this rigorous we will insert the explicit verification that the varying invariant along the fan equals the value required by the Kidder solution at the tail characteristic, using the standard integral expression for the Riemann invariant in the isentropic case. This calculation is algebraic once the cutoff density and the Kidder exponents are substituted and will be included in the revision. revision: yes

  3. Referee: [Numerical results] Numerical section: the simulations are said to support the analytic prediction, yet no grid resolution, convergence study, or quantitative comparison (e.g., L2 error to the claimed closed-form profile) is reported, leaving open whether the observed suppression is due to the exact analytic mechanism or numerical dissipation.

    Authors: We accept that the numerical section requires quantitative support. The revised version will state the spatial and temporal resolutions employed, present a mesh-refinement study with observed convergence rates, and report L2 errors between the computed profiles and the analytic self-similar solution at representative times. These additions will demonstrate that the suppression of the implosion is captured independently of numerical dissipation. revision: yes

Circularity Check

0 steps flagged

No circularity: analytic solution constructed independently from Kidder base plus cutoff rarefaction

full rationale

The paper begins from the standard 1D isentropic Euler system, adopts the known Kidder self-similar form only in the core region, imposes an explicit constant-density cutoff at finite radius as initial data, and derives the resulting non-centered rarefaction by matching Riemann invariants across the fan. No equation is shown to reduce to a fitted parameter renamed as a prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via self-citation. The claimed exact analytic solution is therefore self-contained against the external Kidder benchmark and the standard simple-wave theory; the derivation does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard 1D radially symmetric isentropic compressible Euler equations and the self-similar ansatz from Kidder; the constant-density cutoff is introduced as a modeling choice without additional free parameters or new entities stated in the abstract.

axioms (2)
  • standard math The governing equations are the radially symmetric, isentropic, compressible Euler equations in one dimension.
    Standard background equations invoked for the implosion problem.
  • domain assumption Solutions admit a self-similar form as in Kidder's closed-form implosion.
    Inherited modeling assumption required to obtain the analytic solution with cutoff.

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Works this paper leans on

37 extracted references · 1 canonical work pages

  1. [1]

    Inertial con finement by spherical implosion

    Stefano Atzeni and J¨ urgen Meyer-ter Vehn. Inertial con finement by spherical implosion. In The Physics of Inertial Fusion: Beam Plasma Interaction, Hy drodynamics, Hot Dense Mat- ter, pages 47–74. Oxford University Press, 2004

    2004

  2. [2]

    A second-order Godunov-type scheme for compress- ible fluid dynamics

    Matania Ben-Artzi and Joseph Falcovitz. A second-order Godunov-type scheme for compress- ible fluid dynamics. Journal of Computational Physics , 55(1):1–32, 1984

    1984

  3. [3]

    Self-similar solutions to the compressible Euler equations and their instabilities

    Anxo Biasi. Self-similar solutions to the compressible Euler equations and their instabilities. Communications in Nonlinear Science and Numerical Simulat ion, 103:106014, 2021

    2021

  4. [4]

    Bourgeade, Ph Le Floch, and P

    A. Bourgeade, Ph Le Floch, and P. A. Raviart. An asymptoti c expansion for the solution of the generalized Riemann problem. part 2 : application to t he equations of gas dynamics. Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire, 6(6):437–480, 1989

    1989

  5. [5]

    Brenner, Sascha Hilgenfeldt, and Detlef Lohs e

    Michael P. Brenner, Sascha Hilgenfeldt, and Detlef Lohs e. Single-bubble sonoluminescence. Reviews of Modern Physics , 74(2):425–484, 2002

    2002

  6. [6]

    Smooth imploding solutions for 3D compressible fluids

    Tristan Buckmaster, Gonzalo Cao-Labora, and Javier G´ o mez-Serrano. Smooth imploding solutions for 3D compressible fluids. Forum of Mathematics, Pi , 13:e6, 2025

    2025

  7. [7]

    Non-radial im- plosion for compressible Euler and Navier-Stokes in T3 and R3

    Gonzalo Cao-Labora, Javier G´ omez-Serrano, Jia Shi, an d Gigliola Staffilani. Non-radial im- plosion for compressible Euler and Navier-Stokes in T3 and R3. Cambridge Journal of Math- ematics, 13(4):753–885, 2025

    2025

  8. [8]

    Singularity fo rmation for the compressible Euler equations

    Geng Chen, Ronghua Pan, and Shengguo Zhu. Singularity fo rmation for the compressible Euler equations. SIAM Journal on Mathematical Analysis , 49(4):2591–2614, 2017

    2017

  9. [9]

    Compressible flow and Euler’s equations

    Demetrios Christodoulou and Shuang Miao. Compressible flow and Euler’s equations . Surveys of Modern Mathematics. International Press of Boston; Beij ing, Somerville, Massachusetts, 2014

    2014

  10. [10]

    C. M. Dafermos. The second law of thermodynamics and sta bility. Archive for Rational Mechanics and Analysis , 70(2):167–179, 1979

    1979

  11. [11]

    Dafermos

    Constantine M. Dafermos. Hyperbolic conservation laws in continuum physics . Grundlehren der mathematischen Wissenschaften. Springer, Berlin, Ger many, 4th edition, 2016

    2016

  12. [12]

    Uniqueness of solutions to hyperbolic conservation laws

    Ronald DiPerna. Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ. Math. J. , 28(1):137–188, 1979. SELF-SIMILAR IMPLODING SOLUTIONS 25

    1979

  13. [13]

    Lawrence C. Evans. Partial differential equations . Graduate Studies in Mathematics. Amer- ican Mathematical Society, Providence, Rhode Island, seco nd edition, 2022

    2022

  14. [14]

    Giron, Scott D

    Jesse F. Giron, Scott D. Ramsey, and Roy S. Baty. Nemchin ov–Dyson solutions of the two-dimensional axisymmetric inviscid compressible flow e quations. Physics of Fluids , 32(12):127116, 2020

    2020

  15. [15]

    Guderley

    K.G. Guderley. Starke kugelige und zylindrische verdi chtungsst¨ osse in der n¨ ahe des kugelmit- telpunktes bzw. der zylinderachse. Luftfahrtforschung, 19:302–312, 1942

    1942

  16. [16]

    C. Hunter. On the collapse of an empty cavity in water. Journal of Fluid Mechanics , 8(2):241– 263, 1960

    1960

  17. [17]

    Hunter, Jr

    James H. Hunter, Jr. The collapse of interstellar gas cl ouds and the formation of stars. Monthly Notices of the Royal Astronomical Society , 142(4):473–498, 1969

    1969

  18. [18]

    Amplitude b lowup in radial isentropic Euler flow

    Helge Kristian Jenssen and Charis Tsikkou. Amplitude b lowup in radial isentropic Euler flow. SIAM Journal on Applied Mathematics , 80(6):2472–2495, 2020. doi: 10.1137/20M1340241

    work page doi:10.1137/20m1340241 2020

  19. [19]

    R. E. Kidder. Theory of homogeneous isentropic compres sion and its application to laser fusion. Nuclear Fusion, 14(1):53, 1974

    1974

  20. [20]

    R. E. Kidder. Energy gain of laser-compressed pellets: a simple model calculation. Nuclear Fusion, 16(3):405, 1976

    1976

  21. [21]

    Peter D. Lax. Development of singularities of solution s of nonlinear hyperbolic partial differ- ential equations. Journal of Mathematical Physics , 5(5):611–613, 1964

    1964

  22. [22]

    Peter D. Lax. Hyperbolic Systems of Conservation Laws and the Mathematic al Theory of Shock Waves . CBMS-NSF Regional Conference Series in Applied Mathemati cs. Society for Industrial and Applied Mathematics, 1973

    1973

  23. [23]

    Ph Le Floch and P. A. Raviart. An asymptotic expansion fo r the solution of the generalized Riemann problem part I: General theory. Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire, 5(2):179–207, 1988

    1988

  24. [24]

    Description de la formation d’un choc dans le p-systeme

    Marie-Pierre Lebaud. Description de la formation d’un choc dans le p-systeme. App roxima- tion de valeurs propres par des elements finis isoparametriq ues. Thesis, Universit´ e de Rennes I, 1992. Th` ese de doctorat Math´ ematiques et applications Rennes 1 1992 1992REN10094

    1992

  25. [25]

    Numerical Methods for Conservation Laws

    Randall Leveque. Numerical Methods for Conservation Laws . Springer Basel AG, Basel, 1990

    1990

  26. [26]

    The generalized Riemann problem for quasi linear hyperbolic systems of conser- vation laws

    Ta-tsien Li. The generalized Riemann problem for quasi linear hyperbolic systems of conser- vation laws. In Chaohao Gu, Xiaxi Ding, and Chung-Chun Yang, editors, Partial Differential Equations in China , pages 80–103. Springer Netherlands, Dordrecht, 1994

    1994

  27. [27]

    On the implosion of a compressible fluid I: Smooth self-similar inviscid profiles

    Frank Merle, Pierre Rapha¨ el, Igor Rodnianski, and Jer emie Szeftel. On the implosion of a compressible fluid I: Smooth self-similar inviscid profiles . Annals of Mathematics , 196(2):567– 778, 2022

    2022

  28. [28]

    On the implosion of a compressible fluid II: Singularity formation

    Frank Merle, Pierre Rapha¨ el, Igor Rodnianski, and Jer emie Szeftel. On the implosion of a compressible fluid II: Singularity formation. Annals of Mathematics , 196(2):779–889, 2022

    2022

  29. [29]

    Meyer-ter Vehn and C

    J. Meyer-ter Vehn and C. Schalk. Selfsimilar spherical compression waves in gas dynamics. Zeitschrift f¨ ur Naturforschung A, 37(8):954–970, 1982

    1982

  30. [30]

    W. F. Noh. Errors for calculations of strong shocks usin g an artificial viscosity and an artificial heat flux. Journal of Computational Physics , 72(1):78–120, 1987

    1987

  31. [31]

    Scott D. Ramsey. The linear velocity profile class of hyd rodynamics verification test problems. Full Paper LA-UR-11-02766, Los Alamos National Laboratory , 2011

    2011

  32. [32]

    L. I. Sedov. Chapter IV - One-dimensional unsteady moti on of a gas. In L. I. Sedov, editor, Similarity and Dimensional Methods in Mechanics , pages 146–304. Academic Press, 1959

    1959

  33. [33]

    Thomas C. Sideris. Formation of singularities in three -dimensional compressible fluids. Com- munications in Mathematical Physics , 101(4):475–485, 1985

    1985

  34. [34]

    Eleuterio F. Toro. The generalized riemann problem. In Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction , pages 625–653. Springer Berlin Hei- delberg, Berlin, Heidelberg, 2009

    2009

  35. [35]

    von Neumann and R

    J. von Neumann and R. D. Richtmyer. A method for the numer ical calculation of hydrody- namic shocks. Journal of Applied Physics , 21(3):232–237, 1950

    1950

  36. [36]

    P. R. W oodward. Shock-driven implosion of interstella r gas clouds and star formation. The Astrophysical Journal, 207:484, 1976

    1976

  37. [37]

    Formation and construction of a shock wav e for 3-D compressible Euler equa- tions with the spherical initial data

    Huicheng Yin. Formation and construction of a shock wav e for 3-D compressible Euler equa- tions with the spherical initial data. Nagoya Mathematical Journal , 175:125–164, 2004

    2004