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arxiv: 2604.22073 · v1 · submitted 2026-04-23 · 🧮 math.AP

Cumulative Euler flows

Helge Kristian Jenssen This is my paper

Pith reviewed 2026-05-09 20:33 UTC · model grok-4.3

classification 🧮 math.AP
keywords compressible Euler equationsmass accumulationradial affine motionscumulative flowssingularity formationideal gasLagrangian formulationDirac delta density
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The pith

All radial affine cumulative solutions to the compressible Euler equations start with unphysical unbounded far-field velocity or acceleration.

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

The paper investigates whether solutions to the compressible Euler system for an ideal gas can develop mass accumulation in the form of a Dirac delta in the density field, starting from bounded initial data. It restricts attention to radial affine motions, which are derived via a Lagrangian formulation and do allow for cumulative behavior through either inertial effects or adverse pressure gradients. The central finding is that every such affine cumulative solution necessarily begins with unphysical behavior, as the velocity and/or acceleration is unbounded in the far field at the initial time. The work also examines the evolution of characteristics in these flows and provides examples, including some non-affine cases in one dimension. It concludes by considering whether known singular solutions can be modified to produce physically acceptable gas flows that still display accumulation.

Core claim

We show that all affine cumulative solutions necessarily exhibit unphysical behavior due to initially unbounded velocity and/or acceleration in the far-field. The class of radial affine motions includes examples of cumulative behavior with two distinct mechanisms, due to inertial effects or adverse pressure gradients, respectively. However, the analysis demonstrates that this class always produces flows that violate physical requirements at the initial time in the far field. The paper further analyzes the behavior of characteristics in cumulative flows and considers concrete examples, including a class of 1-dimensional non-affine flows, while discussing the possibility of modifying the known

What carries the argument

Radial affine motions obtained via a Lagrangian formulation, which serve as a test class containing cumulative solutions but always forcing unphysical far-field conditions at t=0.

If this is right

  • Cumulative behavior via inertial effects or adverse pressure gradients in affine radial flows is always accompanied by unbounded initial far-field velocity or acceleration.
  • The two identified accumulation mechanisms cannot produce physically acceptable flows within the affine class.
  • Characteristics in cumulative flows exhibit specific propagation patterns that prevent bounded far-field conditions.
  • Non-affine modifications to known examples may be required to achieve accumulation without initial unphysical behavior.
  • In one dimension, certain non-affine flows still display similar cumulative issues tied to unbounded far-field data.

Where Pith is reading between the lines

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

  • If affine motions capture the generic route to mass accumulation, then Dirac-delta densities may be unreachable in physically reasonable Euler flows without external forces.
  • The restriction to radial symmetry suggests that breaking symmetry could allow bounded cumulative solutions, which could be tested by constructing explicit non-radial examples.
  • The far-field unboundedness may be linked to the absence of external forces, implying that forcing terms might regularize the initial data while preserving accumulation.
  • Numerical simulations of the 1D non-affine class could quantify how far the unboundedness extends before becoming negligible at finite distances.

Load-bearing premise

The analysis is restricted to the class of radial affine motions, and the conclusion that cumulative behavior is unphysical relies on this class being representative enough to draw general conclusions about possible singularities.

What would settle it

A single explicit radial affine cumulative solution in which the initial velocity and acceleration remain bounded at large distances would directly contradict the central claim.

Figures

Figures reproduced from arXiv: 2604.22073 by Helge Kristian Jenssen.

Figure 1
Figure 1. Figure 1: Schematic picture of particle trajectories (solid) and 1-characteristics (dashed) in radial affine flow with separation constant λ = 0. The initial density field is constant and the relation n(γ − 1) = 2 is assumed (so that also the 1- characteristics are straight lines); cf. Example 8.1. In the particular case when n(γ − 1) = 2 (e.g., n = 3, γ = 5 3 ) the 1-characteristics (8.6) are straight lines r(t) = … view at source ↗
Figure 2
Figure 2. Figure 2: Schematic picture of particle trajectories (solid) and 1-characteristics (dashed) in radial affine flow with separation constant λ = 0. The initial density field is singular (¯ρ(s) = s −2 ) and n = 3. In this case both particle trajectories and 1-characteristics accumulate at r = 0 at time tc; cf. Example 8.1. 9. Radial affine flows with separation constant λ ̸= 0 For radial affine flows with separation co… view at source ↗
Figure 3
Figure 3. Figure 3: Case (A): Three (α, α˙)-trajectories for (9.6) with λ > 0. Solid dots indicate initial values (with α(0) = 1) and arrows indicate direction of flow as time increases. The parameter values are n = 3, γ = 5 3 , λ = 1 4 . (Aa) If ¯p(s) is defined with ¯p ′ (s) < 0 for all s > 0, then ¯ρ(s) is defined and strictly positive for all s > 0, i.e., the gas fills all of space. However, (9.2) then yields unbounded sp… view at source ↗
Figure 4
Figure 4. Figure 4: Case (B1): Two (α, α˙)-trajectories for (9.6) with λ < 0 < a. Solid dots indicate initial values (with α(0) = 1) and arrows indicate direction of flow as time increases. The parameter values are n = 3, γ = 5 3 , λ = − 15 16 . (B1a) If ˙α(0) > √ a, then the (α, α˙)-trajectory is located in the upper half of the (α, α˙)-plane, and the ‘+’ sign in (9.11) should be used. We have α(t) > 1 and ˙α(t) > √ a for al… view at source ↗
Figure 5
Figure 5. Figure 5: Case (B1): Two (α, α˙)-trajectories for (9.6) with λ < 0, a < 0. Solid dots indicate initial values (with α(0) = 1) and arrows indicate direction of flow as time increases. The parameter values are n = 3, γ = 5 3 , λ = −3 [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Schematic picture of a particle trajectory (solid) and two 1- characteristics (dashed) in Case (B2) with initial pressure profile ¯p(s) = s κ with κ > 0. All particle paths and all 1-characteristics accumulate at the center of mo￾tion at time tc. Any particle trajectory is overtaken by every 1-characteristic with a larger starting radius. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
read the original abstract

We consider the compressible Euler system for ideal gas flow in the absence of any forces except the internal thermodynamic pressure. In this setting, and in dimensions higher 1, it is known that wave-focusing can drive Euler solutions to amplitude blowup in finite time from bounded initial data. In the known cases (self-similar, radial flows \cites{gud,hun_60,jt3,laz,mrrs1,jls}) the primary flow variables are standard functions at time of blowup. It is natural to ask if the Euler system admits even more singular behavior, and specifically whether accumulation of mass, i.e., the appearance of a Dirac delta in the density field, is possible. We consider the class of radial affine motions \cites{mcvittie,sed, kell,sid_2014} which are conveniently obtained via a Lagrangian formulation. This class does include examples of cumulative behavior, and we observe that there are two distinct mechanisms for accumulation, due to inertial effects or adverse pressure gradients, respectively. However, we show that all affine cumulative solutions necessarily exhibit unphysical behavior due to initially unbounded velocity and/or acceleration in the far-field. We also analyze the behavior of characteristics in cumulative flows and consider concrete examples, including a class of 1-dimensional, non-affine flows. Finally, we discuss the possibility of modifying the known examples to obtain physically acceptable gas flows displaying accumulation.

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

2 major / 3 minor

Summary. The paper considers the compressible Euler system for ideal gas in dimensions greater than 1 and asks whether mass accumulation producing a Dirac delta in the density is possible. Restricting attention to radial affine motions obtained via the Lagrangian formulation, it classifies two distinct mechanisms of accumulation (inertial effects and adverse pressure gradients), derives the far-field asymptotics, and proves that every cumulative solution in this class must have initial data with unbounded velocity or acceleration as |x| → ∞. The manuscript also analyzes the behavior of characteristics in cumulative flows, constructs concrete examples including a class of 1D non-affine flows, and discusses whether the known affine examples can be modified to produce physically acceptable accumulating flows.

Significance. If the central claim holds, the work establishes a rigorous obstruction: within the natural class of radial affine motions, any solution that develops a Dirac mass in density necessarily starts from unphysical initial data. This is a useful negative result for the study of Euler singularities, as it rules out this subclass of candidates while providing an explicit classification of accumulation mechanisms and far-field asymptotics. The inclusion of characteristic analysis, explicit 1D non-affine examples, and a discussion of possible modifications supplies concrete material for future investigations. The paper ships clear derivations from the affine ansatz and explicit examples, which are strengths.

major comments (2)
  1. [§4] §4, the main theorem on unbounded far-field behavior: the argument that both inertial and pressure-driven accumulation force unbounded initial velocity or acceleration is load-bearing for the central claim; the derivation should explicitly enumerate all admissible affine parameters that produce a vanishing Jacobian determinant and confirm that no cancellation with the pressure term can restore boundedness at infinity.
  2. [§6] §6, the 1D non-affine example: while the construction is presented as achieving accumulation, the manuscript should include a direct verification that the initial velocity remains bounded (or state the precise sense in which it differs from the affine case) to support the subsequent discussion on modifications; without this, the example risks appearing as an exception rather than an illustration.
minor comments (3)
  1. [Abstract] Abstract: 'higher 1' should read 'higher than 1' for grammatical clarity.
  2. [§2] §2: the Lagrangian formulation of the affine ansatz would benefit from an explicit table listing the matrix A(t), the Jacobian determinant, and the far-field velocity expression to improve readability.
  3. [§5] §5: the characteristic analysis contains several inline equations whose notation for the speed of sound and the radial coordinate could be standardized with the earlier sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive suggestions. We address the two major comments below and will incorporate the requested clarifications into the revised manuscript.

read point-by-point responses

  1. Referee: [§4] §4, the main theorem on unbounded far-field behavior: the argument that both inertial and pressure-driven accumulation force unbounded initial velocity or acceleration is load-bearing for the central claim; the derivation should explicitly enumerate all admissible affine parameters that produce a vanishing Jacobian determinant and confirm that no cancellation with the pressure term can restore boundedness at infinity.

    Authors: We agree that an explicit enumeration of admissible parameters strengthens the presentation of the main theorem. In the revised version we will insert a short subsection (or dedicated paragraph) that lists all admissible choices of the affine constants for which the Jacobian determinant vanishes at a finite time. For each such choice we will recompute the far-field asymptotics of velocity and acceleration, confirming divergence in every case. We will also add a brief argument showing that the pressure contribution cannot cancel the divergence, as the pressure is controlled by the density, which remains locally integrable and decays at spatial infinity in these solutions. revision: yes

  2. Referee: [§6] §6, the 1D non-affine example: while the construction is presented as achieving accumulation, the manuscript should include a direct verification that the initial velocity remains bounded (or state the precise sense in which it differs from the affine case) to support the subsequent discussion on modifications; without this, the example risks appearing as an exception rather than an illustration.

    Authors: We will revise §6 to include an explicit computation of the initial velocity field for the family of 1D non-affine cumulative flows. The calculation will show that the velocity is bounded (in fact, smooth and compactly supported in the concrete examples we construct). This verification will be placed immediately after the construction and will be used to emphasize the contrast with the affine case, thereby supporting the later discussion on possible modifications toward physically acceptable accumulating solutions. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained within affine class; no reduction to inputs by construction

full rationale

The paper restricts attention to the explicitly defined class of radial affine motions obtained from the Lagrangian formulation of the Euler system. Within this class it classifies two accumulation mechanisms, derives the far-field asymptotics of velocity and acceleration directly from the affine ansatz and the Jacobian determinant, and concludes that any cumulative member must be initially unbounded at infinity. These steps follow from the governing ODEs and standard properties of the Euler equations; they do not invoke fitted parameters renamed as predictions, self-citations that bear the central load, or any uniqueness theorem imported from the authors' prior work. The cited references supply only the historical definition of the ansatz class, not the unphysical-behavior claim itself. Consequently the derivation remains independent of its inputs and receives a circularity score of zero.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard compressible Euler system for ideal gas (no external forces) and the existence of the radial affine motion class obtained via Lagrangian coordinates. No free parameters are introduced in the abstract; the unphysical behavior is derived from the structure of the affine ansatz itself.

axioms (2)
  • domain assumption The compressible Euler equations for an ideal gas hold in the absence of external forces.
    Invoked in the opening paragraph as the governing system.
  • domain assumption Radial affine motions form a closed class under the Euler dynamics and can be obtained exactly via Lagrangian formulation.
    Cited references (mcvittie, sed, kell, sid_2014) are used to justify the class; the abstract treats this as given.

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Works this paper leans on

29 extracted references · 19 canonical work pages

  1. [1]

    Atzeni and J

    S. Atzeni and J. Meyer-ter-Vehn,The Physics of Inertial Fusion, International Series of Monographs on Physics, vol. 125, Oxford University Press, Oxford, 2004

    2004

  2. [2]

    Courant and K

    R. Courant and K. O. Friedrichs,Supersonic flow and shock waves, Springer-Verlag, New York, 1976. Reprinting of the 1948 original; Applied Mathematical Sciences, Vol. 21. MR0421279 (54 #9284)

    1976

  3. [3]

    Yinbin Deng, Tai-Ping Liu, Tong Yang, and Zheng-an Yao,Solutions of Euler-Poisson equations for gaseous stars, Arch. Ration. Mech. Anal.164(2002), no. 3, 261–285, DOI 10.1007/s00205-002-0209-6. MR1930393

    work page doi:10.1007/s00205-002-0209-6 2002

  4. [4]

    Yinbin Deng, Jianlin Xiang, and Tong Yang,Blowup phenomena of solutions to Euler-Poisson equations, J. Math. Anal. Appl.286(2003), no. 1, 295–306, DOI 10.1016/S0022-247X(03)00487-6. MR2009638

    work page doi:10.1016/s0022-247x(03)00487-6 2003

  5. [5]

    Chun-Chieh Fu and Song-Sun Lin,On the critical mass of the collapse of a gaseous star in spherically symmetric and isentropic motion, Japan J. Indust. Appl. Math.15(1998), no. 3, 461–469, DOI 10.1007/BF03167322. MR1651739

    work page doi:10.1007/bf03167322 1998

  6. [6]

    Pure Appl

    James Glimm,Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math.18 (1965), 697–715. MR0194770 (33 #2976)

    1965

  7. [7]

    Guderley,Starke kugelige und zylindrische Verdichtungsst¨ osse in der N¨ ahe des Kugelmittelpunktes bzw

    G. Guderley,Starke kugelige und zylindrische Verdichtungsst¨ osse in der N¨ ahe des Kugelmittelpunktes bzw. der Zylinderachse, Luftfahrtforschung19(1942), 302–311 (German). MR0008522

    1942

  8. [8]

    Yan Guo, Mahir Hadˇ zi´ c, and Juhi Jang,Continued gravitational collapse for Newtonian stars, Arch. Ration. Mech. Anal.239(2021), no. 1, 431–552, DOI 10.1007/s00205-020-01580-w. MR4198723

    work page doi:10.1007/s00205-020-01580-w 2021

  9. [9]

    Hunter,On the collapse of an empty cavity in water, J

    C. Hunter,On the collapse of an empty cavity in water, J. Fluid Mech.8(1960), 241–263

    1960

  10. [10]

    Juhi Jang, Jiaqi Liu, and Matthew Schrecker,On self-similar converging shock waves, Arch. Ration. Mech. Anal. 249(2025), no. 3, Paper No. 24, 83, DOI 10.1007/s00205-025-02096-x. MR4887543

    work page doi:10.1007/s00205-025-02096-x 2025

  11. [11]

    D410(2020), 132511, 14, DOI 10.1016/j.physd.2020.132511

    Helge Kristian Jenssen and Charis Tsikkou,Multi-d isothermal Euler flow: existence of unbounded radial simi- larity solutions, Phys. D410(2020), 132511, 14, DOI 10.1016/j.physd.2020.132511. MR4091348

    work page doi:10.1016/j.physd.2020.132511 2020

  12. [12]

    ,Amplitude blowup in radial isentropic Euler flow, SIAM J. Appl. Math.80(2020), no. 6, 2472–2495, DOI 10.1137/20M1340241. MR4181105

    work page doi:10.1137/20m1340241 2020

  13. [13]

    1, Paper No

    ,Non-isentropic cavity flow for the multi-d compressible Euler system, Nonlinearity39(2026), no. 1, Paper No. 015012, 23, DOI 10.1088/1361-6544/ae35c3. MR5018421

    work page doi:10.1088/1361-6544/ae35c3 2026

  14. [14]

    J. B. Keller,Spherical, cylindrical and one-dimensional gas flows, Quart. Appl. Math.14(1956), 171–184

    1956

  15. [15]

    Lazarus,Self-similar solutions for converging shocks and collapsing cavities, SIAM J

    Roger B. Lazarus,Self-similar solutions for converging shocks and collapsing cavities, SIAM J. Numer. Anal.18 (1981), no. 2, 316–371

    1981

  16. [16]

    Tai-Ping Liu,Compressible flow with damping and vacuum, Japan J. Indust. Appl. Math.13(1996), no. 1, 25–32, DOI 10.1007/BF03167296. MR1377457

    work page doi:10.1007/bf03167296 1996

  17. [17]

    615–624, DOI 10.1080/00411459208203801

    Tetu Makino,Blowing up solutions of the Euler-Poisson equation for the evolution of gaseous stars, Proceedings of the Fourth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Kyoto, 1991), 1992, pp. 615–624, DOI 10.1080/00411459208203801. MR1194464

    work page doi:10.1080/00411459208203801 1991

  18. [18]

    G. C. McVittie,Spherically symmetric solutions of the equations of gas dynamics, Proc. Roy. Soc. London Ser. A220(1953), 339–355, DOI 10.1098/rspa.1953.0191. MR0059719

    work page doi:10.1098/rspa.1953.0191 1953

  19. [19]

    Frank Merle, Pierre Rapha¨ el, Igor Rodnianski, and Jeremie Szeftel,On the implosion of a compressible fluid I: Smooth self-similar inviscid profiles, Ann. of Math. (2)196(2022), no. 2, 567–778, DOI 10.4007/an- nals.2022.196.2.3. MR4445442

    work page doi:10.4007/an- 2022

  20. [20]

    1, 33–91, DOI 10.1088/1361-6544/abb03b

    Calum Rickard, Mahir Hadˇ zi´ c, and Juhi Jang,Global existence of the nonisentropic compressible Euler equa- tions with vacuum boundary surrounding a variable entropy state, Nonlinearity34(2021), no. 1, 33–91, DOI 10.1088/1361-6544/abb03b. MR4183370

    work page doi:10.1088/1361-6544/abb03b 2021

  21. [21]

    Calum Rickard,The vacuum boundary problem for the spherically symmetric compressible Euler equations with positive density and unbounded entropy, J. Math. Phys.62(2021), no. 2, Paper No. 021504, 27, DOI 10.1063/5.0037656. MR4211850

    work page doi:10.1063/5.0037656 2021

  22. [22]

    P. L. Sachdev, K. T. Joseph, and M. Ejanul Haque,Exact solutions of compressible flow equations with spherical symmetry, Stud. Appl. Math.114(2005), no. 4, 325–342, DOI 10.1111/j.0022-2526.2005.01552.x. MR2131550

    work page doi:10.1111/j.0022-2526.2005.01552.x 2005

  23. [23]

    L. I. Sedov,Similarity and dimensional methods in mechanics, “Mir”, Moscow, 1982. Translated from the Russian by V. I. Kisin. MR693457

    1982

  24. [24]

    Fl¨ ugge), Bd

    James Serrin,Mathematical principles of classical fluid mechanics, Handbuch der Physik (herausgegeben von S. Fl¨ ugge), Bd. 8/1, Str¨ omungsmechanik I (Mitherausgeber C. Truesdell), Springer-Verlag, Berlin, 1959, pp. 125–

    1959

  25. [25]

    MR0108116 (21 #6836b)

  26. [26]

    Michael Sever,Distribution solutions of nonlinear systems of conservation laws, Mem. Amer. Math. Soc.190 (2007), no. 889, viii+163, DOI 10.1090/memo/0889. MR2355635

    work page doi:10.1090/memo/0889 2007

  27. [27]

    Sideris,Spreading of the free boundary of an ideal fluid in a vacuum, J

    Thomas C. Sideris,Spreading of the free boundary of an ideal fluid in a vacuum, J. Differential Equations257 (2014), no. 1, 1–14, DOI 10.1016/j.jde.2014.03.006. MR3197239 25

    work page doi:10.1016/j.jde.2014.03.006 2014

  28. [28]

    ,Global existence and asymptotic behavior of affine motion of 3D ideal fluids surrounded by vacuum, Arch. Ration. Mech. Anal.225(2017), no. 1, 141–176, DOI 10.1007/s00205-017-1106-3. MR3634025

    work page doi:10.1007/s00205-017-1106-3 2017

  29. [29]

    Henri Poincar´ e25(2024), no

    ,Expansion and collapse of spherically symmetric isotropic elastic bodies surrounded by vacuum, Ann. Henri Poincar´ e25(2024), no. 7, 3529–3562, DOI 10.1007/s00023-023-01390-2. MR4761415 H. K. Jenssen, Department of Mathematics, Penn State University, University Park, State College, PA 16802, USA (jenssen@math.psu.edu). 26

    work page doi:10.1007/s00023-023-01390-2 2024