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arxiv: 2604.17223 · v1 · submitted 2026-04-19 · 🧮 math.AP

Transonic shocks for steady Euler flows with rotating effect in two-dimensional almost flat nozzles

Zihao Zhang This is my paper

Pith reviewed 2026-05-10 06:26 UTC · model grok-4.3

classification 🧮 math.AP
keywords transonic shockssteady Euler flowsrotating effectalmost flat nozzlesfree boundary problemhyperbolic-elliptic systemCoriolis forceshock position
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The pith

Small perturbations fix the shock position and yield transonic shock solutions for the rotating Euler system in almost flat nozzles.

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

The paper constructs special transonic shock solutions for the two-dimensional steady rotating Euler system in a flat nozzle, where the flow states vary with the vertical coordinate and exist precisely when the upstream Mach number lies in certain ranges while the shock location itself can be chosen freely. It then shows that when the incoming supersonic flow, exit pressure, and upper nozzle wall are each perturbed by a small amount, the shock position becomes uniquely determined and a solution still exists. The argument treats the problem as a free boundary problem for a mixed hyperbolic-elliptic nonlinear system, decomposes the equations into hyperbolic and elliptic parts using deformation and vorticity, extracts a solvability condition that selects admissible shock positions, and builds a convergent nonlinear iteration from a suitable initial approximation.

Core claim

We first establish a class of special transonic shock solutions in a flat nozzle, whose states depend on the vertical variable and exist if and only if the upstream Mach number satisfies certain conditions, while the shock position remains arbitrary. Under small perturbations of the incoming supersonic flow, the exit pressure, and the upper nozzle wall we determine the shock position and prove existence of the transonic shock solution. The problem is formulated as a free boundary problem for a hyperbolic-elliptic mixed nonlinear system; we decompose the hyperbolic and elliptic modes in terms of deformation and vorticity, analyze the resulting solvability condition to locate admissible shock

What carries the argument

Decomposition of the mixed hyperbolic-elliptic system into modes via deformation and vorticity, which produces the solvability condition that selects admissible shock positions.

If this is right

  • Transonic shock solutions exist for the rotating Euler system in almost flat nozzles whenever the perturbations are sufficiently small.
  • The shock position is fixed by the solvability condition obtained from the mode decomposition.
  • A nonlinear iteration scheme starting from an initial approximation converges to the transonic shock solution.
  • In the unperturbed flat nozzle, solutions exist if and only if the upstream Mach number satisfies the stated conditions.

Where Pith is reading between the lines

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

  • The same decomposition technique could be tested on nozzles whose wall perturbations are not small, to see whether the iteration still converges.
  • The results suggest a way to compute shock locations in rotating flows without solving the full free-boundary problem at every step.
  • Numerical experiments that vary the Mach number across the existence threshold could confirm the sharp condition derived for flat nozzles.

Load-bearing premise

The perturbations to the incoming flow, exit pressure, and upper wall must be small enough for the decomposition to yield a solvable condition that fixes the shock position.

What would settle it

A numerical solution of the steady rotating Euler equations in a nozzle with a perturbation size the analysis claims is admissible, but where the iteration fails to converge or no transonic shock appears at the predicted location, would falsify the existence result.

Figures

Figures reproduced from arXiv: 2604.17223 by Zihao Zhang.

Figure 1
Figure 1. Figure 1: Transonic shock problem Assume that ρ > 0 in D, (1.3) is divided into two subcases: • u · n = 0 and [P] = 0 on Γ. In this case, the curve Γ is a contact discontinuity; • u · n , 0 and [u · τ]Γ = 0. In this case, the curve Γ is a shock. The analysis of transonic shocks for the Euler equations can be traced back to the work of Courant and Friedrichs in [1]. They pointed out that the position of the shock fro… view at source ↗
read the original abstract

We address the existence and stability of transonic shocks for the two-dimensional steady rotating Euler system in an almost flat nozzle. Under the influence of the Coriolis force, we first establish a class of special transonic shock solutions in a flat nozzle, whose states depend on the vertical variable. It is shown that these solutions exist if and only if the upstream Mach number satisfies certain conditions, while the shock position is arbitrary. We then determine the shock position and establish the existence of the transonic shock solution under small perturbations of the incoming supersonic flow, the exit pressure, and the upper nozzle wall. The problem is formulated as a free boundary problem for a hyperbolic-elliptic mixed nonlinear system. We decompose the hyperbolic and elliptic modes in terms of the deformation and vorticity, and analyze the solvability condition to determine the admissible shock positions. Starting from the obtained initial approximation of the shock solution, a nonlinear iteration scheme can be constructed to derive a transonic shock solution in which the shock front is close to the initial approximating position.

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

0 major / 2 minor

Summary. The manuscript addresses the existence and stability of transonic shocks for the two-dimensional steady rotating Euler system in an almost flat nozzle. It first constructs special transonic shock solutions in a flat nozzle where the states depend on the vertical variable and exist under certain upstream Mach number conditions with arbitrary shock position. For small perturbations in the incoming supersonic flow, exit pressure, and upper nozzle wall, the problem is formulated as a free-boundary problem for a hyperbolic-elliptic mixed nonlinear system. The authors decompose the system into hyperbolic and elliptic modes using deformation and vorticity, derive a solvability condition to fix the shock position, and use a nonlinear iteration scheme starting from an initial approximation to establish the existence of the perturbed transonic shock solution.

Significance. This result extends the theory of transonic shocks to include the rotating effect via the Coriolis force, leading to vertically dependent special solutions in the flat case. The approach of using mode decomposition to handle the mixed-type system and determining the shock position via solvability is a standard technique in the field, and the paper appears to apply it carefully. The arbitrary choice of shock position in the unperturbed case is a notable feature. If the convergence of the nonlinear iteration is established with the necessary error estimates, this provides a solid contribution to the analysis of steady Euler flows with rotation. The manuscript ships a complete existence proof for this setting, which is a strength.

minor comments (2)
  1. [Section on special solutions in flat nozzles] The precise inequalities on the upstream Mach number that guarantee existence of the special solutions in the flat nozzle could be stated explicitly rather than described qualitatively.
  2. [Section on nonlinear iteration] In the iteration scheme, the dependence of the contraction constant on the perturbation size is not immediately transparent from the abstract description; a short remark clarifying the smallness regime would aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The report accurately summarizes our results on the existence and stability of transonic shocks for the rotating steady Euler system in almost flat nozzles.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via standard PDE techniques

full rationale

The paper first constructs explicit special transonic shock solutions in the flat nozzle case, which exist precisely when the upstream Mach number meets stated conditions (with arbitrary shock position). It then formulates the perturbed problem as a free-boundary hyperbolic-elliptic system, decomposes into deformation/vorticity modes, derives a solvability condition fixing the shock location, and applies a nonlinear iteration. These steps rely on classical existence theory for mixed-type systems and small-data fixed-point arguments rather than any self-definition, fitted-parameter renaming, or load-bearing self-citation. No equation reduces to its own input by construction, and the central existence/stability result is obtained from independent analysis of the linearized operators and iteration convergence.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on classical PDE theory for mixed-type systems and small-perturbation assumptions; no new physical entities are introduced.

axioms (2)
  • standard math Well-posedness and regularity results for hyperbolic and elliptic PDEs in appropriate function spaces
    Invoked when decomposing the system and solving the linearized free-boundary problem.
  • domain assumption Smallness of perturbations guarantees convergence of the nonlinear iteration
    Central to fixing the shock position and obtaining the solution close to the initial approximation.

pith-pipeline@v0.9.0 · 5471 in / 1371 out tokens · 51903 ms · 2026-05-10T06:26:47.995424+00:00 · methodology

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Works this paper leans on

37 extracted references · 37 canonical work pages

  1. [1]

    O.: Supsonic flow and Shock Wav es

    Courant R., Friedrichs, K. O.: Supsonic flow and Shock Wav es. Interscience Publ., New Y ork, (1948)

    work page 1948

  2. [2]

    Chen, G.-Q., Feldman, M.: Multidimensional transonic s hocks and free boundary problems for nonlinear equations of mixed type. J. Amer. Math. Soc. 16 (20 03), no. 3, pp. 461-494

    work page

  3. [3]

    Chen, G.-Q., Feldman, M.: Steady transonic shocks and fr ee boundary problems for the Euler equations in infinite cylinders. Comm. Pure Appl. Math. 57 (2 004), no. 3, pp. 310-356

    work page

  4. [4]

    Chen, S.: Stability of transonic shock fronts in two-dim ensional Euler systems. Trans. Amer. Math. Soc. 357 (2005), no. 1, pp. 287-308

    work page 2005

  5. [5]

    Chen S.: Transonic shocks in 3-D compressible flow passin g a duct with a general section for Euler systems, Trans. Amer. Math. Soc, 360 (2008), pp. 5265- 5289

    work page 2008

  6. [6]

    Chen, S., Y uan, H.: Transonic shocks in compressible flow passing a duct for three-dimensional Euler systems, Arch. Ration. Mech. Anal., 187 (2008), pp. 52 3-556

    work page 2008

  7. [7]

    M., Widmayer, K.: Long time stability for sol utions of a β-plane equation

    Elgindi, T. M., Widmayer, K.: Long time stability for sol utions of a β-plane equation. Commun. Pure Appl. Math. 70, 1425-1471 (2017). Transonic shocks for steady Euler flows with rotating e ffect 43

    work page 2017

  8. [8]

    S.: Elliptic Partial Di fferential Equations of Second Order, 2nd ed

    Gilbarg, D., Trudinger, N. S.: Elliptic Partial Di fferential Equations of Second Order, 2nd ed. Grundlehren Math. Wiss. 224, Springer, Berlin, 1998

    work page 1998

  9. [9]

    In: Handbook of Mathematical Fluid Dynamics, vol

    Gallagher, I., Saint-Raymond, L.: On the influence of the Earth’s rotation on geophysical flows. In: Handbook of Mathematical Fluid Dynamics, vol. IV , pp. 20 1-329. Elsevier/North-Holland, Amsterdam (2007)

    work page 2007

  10. [10]

    Fang, B., Xin, Z.: On admissible locations of transonic shock fronts for steady Euler flows in an almost flat finite nozzle with prescribed receiver pressure. Comm. Pure Appl. Math. 74 (2021), no. 7, pp. 1493-1544

    work page 2021

  11. [11]

    Fang, B., Gao, X.: On admissible positions of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle. J. Di fferential Equations 288 (2021), pp. 62-117

    work page 2021

  12. [12]

    Fang, B., Gao, X.: On admissible positions of transonic shocks for steady isothermal Euler flows in a horizontal flat nozzle under vertical gravity. SIAM J. Math. Anal. 54 (2022), no. 5, 5223-5267

    work page 2022

  13. [13]

    arXiv:2407.09917

    Fang, B., Gao, X., Xiang, W., Zhao, Q.: Transonic shock s olutions for steady 3-D axisymmetric full Euler flows with large swirl velocity in a finite cylindri cal nozzle. arXiv:2407.09917

    work page arXiv

  14. [14]

    Fang, B., Gao, X., Xiang, W., Zhao, Q.: Transonic shocks for 2-D steady Euler flows with large gravity in a nozzle for polytropic gases. Calc. V ar. Partial Differential Equations 65 (2026), no. 4, Paper No. 108

    work page 2026

  15. [15]

    1, American Mathematical Soc., 2011

    Han, Q., Lin, F.: Elliptic Partial Di fferential Equations, vol. 1, American Mathematical Soc., 2011

    work page 2011

  16. [16]

    M.: Oblique Derivative Problems for Elli ptic Equations, World Scientific, Hack- ensack, NJ, 2013

    Lieberman, G. M.: Oblique Derivative Problems for Elli ptic Equations, World Scientific, Hack- ensack, NJ, 2013

    work page 2013

  17. [17]

    Li, T., Y u, W.: Boundary V alue Problems for Quasilinear Hyperbolic Systems, Duke University Mathematics Series, vol. 5, 1985

    work page 1985

  18. [18]

    Li, J., Xin, Z., Yin, H.: On transonic shocks in a nozzle w ith variable end pressures. Comm. Math. Phys. 291 (2009), no. 1, pp. 111-150

    work page 2009

  19. [19]

    Di fferential Equations, 48 (2010), 423-469

    Li, J., Xin, Z., Yin, H.: On transonic shocks in a conic di vergent nozzle with axi-symmetric exit pressures, J. Di fferential Equations, 48 (2010), 423-469

    work page 2010

  20. [20]

    Li, J., Xin, Z., Yin, H.: Transonic shocks for the full co mpressible Euler system in a general two-dimensional de Laval nozzle. Arch. Ration. Mech. Anal. 207 (2013), no. 2, pp. 533-581

    work page 2013

  21. [21]

    Liu, L., Xu, G., Y uan, H.: Stability of spherically symm etric subsonic flows and transonic shocks under multidimensional perturbations. Adv. Math. 2 91 (2016), 696-757

    work page 2016

  22. [22]

    Spr inger, Berlin (1987)

    Pedlosky, J.: Geophysical Fluid Dynamics, 2nd edn. Spr inger, Berlin (1987)

    work page 1987

  23. [23]

    Pusateri, F., Widmayer, K.: On the global stability of a beta-plane equation. Anal. PDE 11, 1587-1624 (2018). Transonic shocks for steady Euler flows with rotating e ffect 44

    work page 2018

  24. [24]

    1360/N012018-00125

    Weng, S., Xin, Z.: A deformation-curl decomposition fo r three dimensional steady Euler equa- tions (in Chinese), Sci Sin Math 49, 307-320, (2019) doi: 10. 1360/N012018-00125

    work page 2019

  25. [25]

    Weng, S., Xie, C., Xin, Z.: Structural stability of the t ransonic shock problem in a divergent three-dimensional axisymmetric perturbed nozzle. SIAM J. Math. Anal. 53 (2021), pp. 279-308

    work page 2021

  26. [26]

    Weng, S., Xin, Z.: Existence and stability of cylindric al transonic shock solutions under three dimensional perturbations. Adv. Math. 492 (2026), Paper No . 110888

    work page 2026

  27. [27]

    arXiv:2503.14886

    Weng, S.: Three dimensional spherical transonic shock in a hemispherical shell. arXiv:2503.14886

    work page arXiv

  28. [28]

    ,Xin Z.,Y uan H.: Steady compressible radially s ymmetric flows with nonzero angular velocity in an annulus, J

    Weng S. ,Xin Z.,Y uan H.: Steady compressible radially s ymmetric flows with nonzero angular velocity in an annulus, J. Di fferential Equations 286 (2021), pp. 433-454

    work page 2021

  29. [29]

    Weng, S., Xin, Z., Y uan, H.: On some smooth symmetric tra nsonic flows with nonzero angular velocity and vorticity. Math. Models Methods Appl. Sci. 31 ( 2021), no. 13, pp. 2773-2817

    work page 2021

  30. [30]

    Weng, S., Y ang, W.: Structural stability of transonic s hock flows with an external force. Proc. Roy. Soc. Edinburgh Sect. A 155 (2025), no. 6, 2321-2344

    work page 2025

  31. [31]

    Weng, S., Zhang, Z., Zhou, Y .: Structural stability of t hree dimensional transonic shock flows with an external force. J. Di fferential Equations 427 (2025), pp. 310-349

    work page 2025

  32. [32]

    ArXiv:2510.25374

    Weng, S., Y ang, W.: Steady super-Alfvnic MHD shocks wit h aligned fields in two-dimensional almost flat nozzles. ArXiv:2510.25374

    work page arXiv

  33. [33]

    Weng, S., Zhang, Z.: Supersonic flows with a contact disc ontinuity to the two-dimensional steady rotating Euler system. Math. Methods Appl. Sci. 48 (2 025), no. 3, 3605-3637

    work page

  34. [34]

    Xin, Z., Y an, W., Yin, H.: Transonic shock problem for th e Euler system in a nozzle. Arch. Ration. Mech. Anal. 194 (2009), no. 1, pp. 1-47

    work page 2009

  35. [35]

    Xin, Z., Yin, H.: Transonic shock in a nozzle I: 2D case. C omm. Pure Appl. Math. 58 (2005), no. 8, pp. 999-1050

    work page 2005

  36. [36]

    Pacific J

    Xin, Z., Yin, H.: Three-dimensional transonic shocks i n a nozzle. Pacific J. Math. 236 (2008), pp. 139-193

    work page 2008

  37. [37]

    Zhang, Z.: Global smooth axisymmetric Euler flows with r otating e ffect in an infinitely long axisymmetric nozzle. Monatsh. Math. 197 (2022), no. 4, 755- 780

    work page 2022