Pith. sign in

REVIEW 3 major objections 4 minor 31 references

3-D axisymmetric transonic shock solutions of the full Euler system in divergent nozzles

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Radial transonic shock solutions in divergent nozzles are stable under small axisymmetric perturbations of the incoming supersonic flow and the exit pressure, including perturbations with nonzero angular momentum.

desk verdict A substantial stability theorem for 3-D axisymmetric transonic shocks with swirl, honestly proved for the restricted data class B = B0 on the entrance, but Remark 2.13 advertises a generality the proof does not deliver. read the letter →

arxiv 1908.04945 v3 pith:6INAYOII submitted 2019-08-14 math.AP math-phmath.MP

Yong Park This is my paper · ORCID
classification math.APmath-phmath.MP MSC 35A0135A0235J5735J6235M1035Q3135R3576H05
keywords transonicshockfullEulersystemstreamfunctionfreeboundaryproblemaxisymmetricflowdivergentnozzleexitpressurenonzeroangularmomentum
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 establishes that radial transonic shocks in divergent nozzles are structurally stable: if the incoming supersonic flow and the exit pressure are changed by a small axisymmetric perturbation, there is a nearby shock surface and downstream subsonic solution of the steady full Euler equations, unique in a natural weighted Hölder class, and the perturbation may carry nonzero angular momentum. The proof works in a stream-function formulation that avoids the singularity at the symmetry axis, and it treats the shock's initial radius and its shape as two separate unknowns determined by two different mechanisms. This matters because previous full-Euler results either excluded angular momentum, required small nozzle opening angle, or imposed additional conditions to pin down the shock location.

What carries the argument

The argument runs through a stream-function formulation of the axisymmetric full Euler system written with the vector-potential stream function $\Phi e_\phi$, the swirl variable $L=2\pi r\sin\theta\,u_\phi$, the entropy $S$, and the Bernoulli invariant $B$. The divergence equation is built into the identity $\mathrm{div}(\nabla\times(\Phi e_\phi))=0$, and the remaining equations become one singular elliptic equation for $\Phi$ coupled to two transport equations for $L$ and $S$. The singularity at the axis is resolved by converting the singular scalar equation into an elliptic system of the form $\mathrm{div}(A D(\Psi e_\phi))-d\,\Psi e_\phi=-F$, whose coefficients are positive-definite because the downstream reference flow is subsonic; existence and regularity follow from Lax-Milgram, freezing of coefficients, and a reflection argument at the two perpendicular corners. The transport equations are solved by following level surfaces of the modified stream function $V=2\pi r\sin\theta(\Phi_0+\Psi)$, giving solutions of the form $Q=Q_{\mathrm{en}}(L)$. The shock location is split into two unknowns: the initial shock radius is fixed by a compatibility equation whose solvability rests on monotonicity of the downstream entropy in the shock position, and the shock shape is fixed by inverting the Fréchet derivative of the tangential Rankine-Hugoniot condition, using eigenfunctions of the associated Legendre problem of type $m=1$ on the interval $(0,\theta_1)$.

What would settle it

Evaluate the derivative in Lemma 2.10, $\frac{d}{dt}\bar S|_{D_t^+}(t;t)$, for a strongly divergent nozzle with $\gamma$ close to $1$; a nonpositive value at some $t\in(r_0,r_1)$ would break the shock-position adjustment mechanism. Alternatively, take a small entrance perturbation with $B\ne B_0$ and try to close the fixed-point scheme; if no solution satisfying estimate (2.5.14) exists, the unproved claim in Remark 2.13 fails.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2.17: for any Hölder exponent $\alpha\in(2/3,1)$ there is a $\sigma_2>0$ such that whenever the axisymmetric incoming supersonic flow differs from the fixed radial background by $\sigma$ in $C^{2,\alpha}$, with the Bernoulli invariant fixed to $B_0$ on the entrance, and the exit pressure differs from $p_c$ by $\sigma$ in the appropriate weighted norm, Problem 1 has a unique axisymmetric transonic shock solution with shock front $r=f(\theta)$ and downstream density, velocity and pressure satisfying estimate (2.5.14), so all deviations from the radial shock are controlled by $C\sigma$. The proof first establishes the equivalent stream-function Problem 2, Theorem 2.16, and then recovers the physical variables through an implicit relation that determines the density from the stream-function unknowns. In the author's terms, this is structural stability of radial transonic shock solutions in divergent nozzles, with no restriction on the nozzle tip angle and no assumption that the incoming supersonic solution itself be radial.

Load-bearing premise

The load-bearing premise is that the incoming and downstream flows share the exact Bernoulli constant $B_0$ of the reference radial flow; the paper asserts in Remark 2.13 that this restriction can be dropped without changing the result, but supplies no proof, so if that assertion fails the theorem covers only entrance perturbations with $B=B_0$.

Editorial extensions

If this is right

  • For every sufficiently small $\sigma$, Problem 1 has a solution and it is unique in the class satisfying (2.5.14): a nearby shock front and nearby subsonic flow exist, with all deviations controlled by $C\sigma$.
  • The shock location is determined by the exit pressure alone, with no assumption that the shock passes through a prescribed wall point; the initial shock radius and the shock shape are fixed by two different conditions.
  • Perturbations with nonzero angular momentum are admitted: the swirl variable $L$ is transported along the streamlines, and the resulting velocity field remains regular across the shock.
  • The result covers nozzles of arbitrary opening angle $\theta_1<\pi$ and flows with $C^{1,\alpha}$ interior and $C^\alpha$ boundary regularity, so general axisymmetric nozzle walls are within reach.

Reading between the lines

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

  • Beyond the paper: if the assertion in Remark 2.13 can be upgraded to a proof, the theorem becomes full structural stability under arbitrary small axisymmetric entrance perturbations; currently the proven statement is conditional on the entrance Bernoulli invariant being fixed to $B_0$.
  • Beyond the paper: the two-step scheme, mass-flux compatibility for the initial shock radius and tangential Rankine-Hugoniot for the shock shape, suggests a numerical shock-fitting algorithm that alternates these two updates instead of solving the full free-boundary problem monolithically.
  • Beyond the paper: the completeness proof for the associated Legendre problem of type $m=1$ on a general interval is a technical tool that likely transfers to other axisymmetric free-boundary elliptic systems with conical corners.
  • Beyond the paper: the lower bound $\alpha>2/3$ appears tied to the corner regularity obtained by the reflection argument; testing whether the Hölder exponent can be lowered with a different corner treatment would clarify the optimal regularity threshold.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies 3-D axisymmetric transonic shock solutions of the steady full Euler system in divergent nozzles. The main result, Theorem 2.17 (via Theorem 2.16 for a stream-function formulation), asserts that a given radial transonic shock solution is structurally stable under sufficiently small axisymmetric perturbations of the incoming supersonic radial flow and of the exit pressure, including perturbations with nonzero angular momentum, with no restriction on the nozzle tip angle and without an S-condition. The proof recasts the downstream subsonic problem in a vector-potential stream-function formulation, splits the free boundary into an initial shock position and a shock shape, solves a Pseudo Free Boundary Problem by a Schauder fixed point argument after linearization, and then determines the shock shape by a weak implicit function theorem. The paper also resolves the axis singularity of the stream-function formulation by solving the singular elliptic equation as an elliptic system and proves an orthogonal completeness statement for an associated Legendre problem.

Significance. If the main theorem is correct, this is a significant contribution to the transonic-shock stability literature: it extends prior 3-D axisymmetric results to nonzero swirl, removes small-tip-angle restrictions, and avoids the S-condition used in some earlier general 3-D work. The manuscript contains substantial original technical machinery: the vector-potential stream-function reformulation, the monotonicity argument (Lemma 2.10) that fixes the initial shock position through the exit-pressure solvability condition, and an implicit-function scheme for the shock shape that avoids the nonlocal elliptic equation appearing in earlier iteration schemes. These are real strengths and make the paper worth serious consideration. However, the advertised range of admissible incoming perturbations is narrower than what is proved unless Remark 2.13 is supplied with a proof, and several regularity statements that are load-bearing for the fixed-point argument are either omitted or delegated to previous work.

major comments (3)
  1. [§2.3–§2.5, Problem 1, Eq. (2.3.2) and Remark 2.13] Problem 1 fixes the Bernoulli invariant exactly, B = B0 on Γen, and §2.5 uses this assumption to impose B = B0 throughout the downstream region and to reduce the Rankine-Hugoniot conditions, replacing [B]Γ = 0 in (2.4.20) by B = B0 in N+f. Every subsequent object—Problem 2, Problem 3, the linearized system (3.1.7)–(3.1.14), the Fréchet derivative in Lemma 4.2, and the implicit-function argument in §4—depends on that reduction. Remark 2.13 asserts that the result is unchanged for a general perturbation of (ρ−0, u−0er, p−0), but no transport equation for B−B0, no treatment of the Rankine-Hugoniot condition [B]Γ = 0, and no modified fixed-point formulation are provided. As written, Theorem 2.17 covers only the codimension-one class of entrance data satisfying (2.3.2), which is weaker than the abstract's claim of stability under small perturbations of an incoming radial supersonic flow. This is a load-bearing gap: please either prove Remark 2.13 or revise the abstract and introduction to state the theorem with the restriction B = B0.
  2. [§3.2, Corollary 3.12 and Lemma 3.13] Corollary 3.12 is stated with the sentence 'The following Corollary is obtained from Lemma 3.11 in the same way that Corollary 3.11 is obtained from Lemma 3.10 in [17]. We omit the proof.' This corollary is then used directly in the proofs of Lemma 3.9 and Lemma 3.10 to obtain the Cβ and C1,α regularity up to the corners Γf ∩ Γ+w and Γ+w ∩ Γex. Lemma 3.13, which upgrades this to C^{2,α}_{(-1-α,Γ+w)}(N+f), is also stated without proof, with only a reference to a scaling argument from [2] and Theorem 5.21 of [16]. These regularity results are essential for the Schauder fixed-point argument in Proposition 3.1 and for the Fréchet differentiability in Lemma 4.2. The manuscript should either provide the full proofs, or give precise theorem statements in the cited works with a verification that all hypotheses (especially the corner-reflection condition and the structure of the boundary data) are met in this setting.
  3. [§3.4, proof of Proposition 3.1, existence step] The existence proof of Proposition 3.1 constructs a Schauder map on the compact convex set P(M1) and then states that continuity of the map J follows by 'the standard argument.' Given that J involves solving the transport equations (B′), the linear elliptic problem (3.4.25)–(3.4.26), and the intermediate-value step defining f(0), continuity is not entirely immediate and depends on the uniqueness estimates for Problems 3.1 and on the regularity of the elliptic solver. Since the entire fixed-point step rests on this continuity assertion, a short proof or a precise reference for the continuity of this composition should be supplied.
minor comments (4)
  1. [Throughout] There are numerous typographical errors, including 'incomming' for 'incoming', 'Prolem' for 'Problem', 'Lemam' for 'Lemma', and 'funtion' for 'function'. These should be corrected in a revision.
  2. [Abstract and Introduction] The abstract and the introduction state the result for 'small perturbations of an incoming radial supersonic flow' without mentioning the condition B = B0 on the entrance. Until Remark 2.13 is proved, the statement of the main theorem should explicitly include this restriction.
  3. [§3.4, after (3.4.59)–(3.4.60)] In the uniqueness part of Proposition 3.1, the estimates labelled (3.4.59) and (3.4.60) are cited before they are stated; renumbering or moving these displayed estimates would improve readability.
  4. [§4.2] The uniqueness proof of Theorem 2.16 refers to 'the arguments in Step 1 in the proof of Lemma 4.2' for the unique solvability of (4.1.8)–(4.1.12) at low regularity; since that system is derived by a formal limiting process, a direct statement of the low-regularity well-posedness would help the reader verify the contraction argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: shock position and shape are solved from the free-boundary equations by IVT and Fréchet-derivative inversion, not fitted; B=B0 is a genuine hypothesis, and Remark 2.13 only leaves an unsupported generalization.

full rationale

Walking the derivation chain, no step reduces the claimed result to its own input. The background radial family in §2.2 is solved from the ODEs (2.2.3)-(2.2.10); Lemma 2.10 derives entropy monotonicity directly from g'(x)>0 and the monotonicity of the upstream Mach number, and Proposition 2.11 uses it to select the background shock rs by the exit-pressure equation. In the nonlinear argument, the initial shock position is not a fitted parameter: Proposition 3.1 solves the scalar equation (3.4.15) for f(0) by the intermediate value theorem, using the strict monotonicity of (L), and then closes the Pseudo Free Boundary Problem with a Schauder fixed point. The shock shape is determined in §4 by the map A in (4.1.1), whose zero enforces the missing tangential Rankine-Hugoniot condition; the Fréchet derivative is computed in Lemma 4.2 and shown invertible in Lemma 4.4 via eigenfunction expansions and a Fredholm argument, with the needed orthogonal completeness proved in Lemma 4.3. Thus no equation is equivalent by construction to the target estimate, and no parameter is renamed as a prediction. The only caveat is scope, not circularity: Problem 1 explicitly fixes B=B0 on Γen in (2.3.2), and §2.5 uses this to impose B=B0 throughout N+f, while Remark 2.13 asserts without proof that the theorem would be unchanged for a general perturbation of (ρ−0,u−0er,p−0). That remark is an unsupported generality claim; the theorem as stated contains (2.3.2), so the caveat affects the advertised breadth but does not make the proof circular. Cited tools such as the weak implicit function theorem from [3] are external analytic facts, not the target result, and the paper proves the auxiliary spectral fact in Lemma 4.3 rather than importing it.

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

The proof introduces no fitted constants and no new physical entities. The main extra burden is the Bernoulli constraint B = B0, which is a stated assumption on the data class rather than a derived consequence. The vector potential stream function is a mathematical reformulation, not an invented physical entity. All other assumptions are standard background theory or the explicit physical setup.

assumptions (5)
  • standard math Standard functional analysis and elliptic regularity theorems: Lax-Milgram, Schauder fixed point theorem, Fredholm alternative, spectral theorem, maximum principle, and interior/boundary regularity for elliptic systems.
    Used throughout Sections 3 and 4 to solve the linearized elliptic system (Lemma 3.7), obtain regularity (Lemmas 3.9, 3.10, 3.13), apply a fixed point argument (Proposition 3.1), and prove invertibility of the linearized shock-location map (Lemma 4.4).
  • domain assumption The flow is a steady inviscid ideal polytropic gas governed by the full Euler system (2.1.1), in an axisymmetric divergent nozzle (2.2.1), with slip boundary conditions and an exit pressure condition.
    This is the physical and geometric setting of Problem 1; the whole analysis is carried out under these hypotheses.
  • domain assumption A supersonic axisymmetric solution (rho-, u-, p-) of the full Euler system exists in the nozzle and is a small perturbation of the radial supersonic solution, as expressed in (2.3.3).
    Remark 2.12 explicitly says this existence is assumed. The proof needs the incoming solution to define the shock boundary data and the Rankine-Hugoniot conditions.
  • ad hoc to paper The Bernoulli invariant B is fixed exactly to B0 on the entrance, equation (2.3.2), and is imposed downstream as B = B0.
    This restriction is used in Section 2.5 to reduce (2.1.1) and (2.1.3). Remark 2.13 claims the restriction can be removed without changing the result, but no proof of that removal is given. The theorem as stated depends on this extra constraint on the admissible perturbations.
  • ad hoc to paper The admissible shock shape functions satisfy f_s'(0) = f_s'(theta1) = 0, and the shock front f is assumed to meet the wall with f'(theta1) = 0 in the elliptic regularity machinery.
    Problem 3 and Lemma 3.5 impose this endpoint condition. The reflection argument for corner regularity in Lemma 3.11 relies on perpendicular corners. This restricts the class of allowed shock perturbations.

how reviews work

0 comments
Cite this review

Pith. "Pith review of 3-D axisymmetric transonic shock solutions of the full Euler system in divergent nozzles." pith.science (2026). https://pith.science/paper/6INAYOII

@misc{pith2026190804945,
  author       = {Pith},
  title        = {Pith review of: 3-D axisymmetric transonic shock solutions of the full Euler system in divergent nozzles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6INAYOII}},
  note         = {Machine review of arXiv:1908.04945}
}
read the original abstract

We establish the stability of 3-D axisymmetric transonic shock solutions of the steady full Euler system in divergent nozzles under small perturbations of an incoming radial supersonic flow and a constant pressure at the exit of the nozzles. To study 3-D axisymmetric transonic shock solutions of the full Euler system, we use a stream function formulation of the full Euler system for a 3-D axisymmetric flow. We resolve the singularity issue arising in stream function formulations of the full Euler system for a 3-D axisymmetric flow. We develop a new scheme to determine a shock location of a transonic shock solution of the steady full Euler system based on the stream function formulation.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [17]

    1, Courant Institute of Mathematical Sci ences, New York; American Mathematical Society, Providence, RI, 2011

    Qing Han and Fanghua Lin, Elliptic partial differential equations , second ed., Courant Lecture Notes in Mathematics, vol. 1, Courant Institute of Mathematical Sci ences, New York; American Mathematical Society, Providence, RI, 2011. MR 2777537

    work page 2011

  2. [2]

    Myoungjean Bae, Ben Duan, and Chunjing Xie, Subsonic flow for the multidimensional Euler-Poisson sys- tem, Arch. Ration. Mech. Anal. 220 (2016), no. 1, 155–191. MR 3458161

    work page 2016

  3. [16]

    Scuola Normale Superiore di Pisa (Nuo va Serie) [Lecture Notes

    Mariano Giaquinta and Luca Martinazzi, An introduction to the regularity theory for elliptic syste ms, har- monic maps and minimal graphs , second ed., Appunti. Scuola Normale Superiore di Pisa (Nuo va Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Serie s)], vol. 11, Edizioni della Normale, Pisa, 2012. MR 3099262

    work page 2012

  4. [1]

    Scuola Normale Superiore di Pisa (Nuova Serie) [L ecture Notes

    Luigi Ambrosio, Alessandro Carlotto, and Annalisa Mass accesi, Lectures on elliptic partial differential equa- tions, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [L ecture Notes. Scuola Normale Superiore di Pisa (New Series)]], vol. 18, Edizioni della Normale, Pis a, 2018

    work page 2018

  5. [3]

    Ra- tion

    Myoungjean Bae and Mikhail Feldman, Transonic shocks in multidimensional divergent nozzles , Arch. Ra- tion. Mech. Anal. 201 (2011), no. 3, 777–840. MR 2824466

    work page 2011

  6. [4]

    Myoungjean Bae and Shangkun Weng, 3-D axisymmetric subsonic flows with nonzero swirl for the co mpress- ible Euler-Poisson system , Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire35 (2018), no. 1, 161–186. MR 3739930

    work page 2018

  7. [5]

    Gui-Qiang Chen, Jun Chen, and Mikhail Feldman, Transonic shocks and free boundary problems for the full Euler equations in infinite nozzles , J. Math. Pures Appl. (9) 88 (2007), no. 2, 191–218. MR 2348768

    work page 2007

  8. [6]

    Differential Equations 229 (2006), no

    Gui-Qiang Chen, Jun Chen, and Kyungwoo Song, Transonic nozzle flows and free boundary problems for the full Euler equations , J. Differential Equations 229 (2006), no. 1, 92–120. MR 2265619

    work page 2006

Show all 31 references
  1. [7]

    Gui-Qiang Chen and Mikhail Feldman, Multidimensional transonic shocks and free boundary probl ems for nonlinear equations of mixed type , J. Amer. Math. Soc. 16 (2003), no. 3, 461–494. MR 1969202 [8] , Steady transonic shocks and free boundary problems for the E uler equations...

    2003

  2. [9]

    , Existence and stability of multidimensional transonic flow s through an infinite nozzle of arbitrary cross-sections, Arch. Ration. Mech. Anal. 184 (2007), no. 2, 185–242. MR 2299761

    2007

  3. [10]

    Chen and Hairong Yuan, Local uniqueness of steady spherical transonic shock-fron ts for the three-dimensional full Euler equations , Commun

    Gui-Qiang G. Chen and Hairong Yuan, Local uniqueness of steady spherical transonic shock-fron ts for the three-dimensional full Euler equations , Commun. Pure Appl. Anal. 12 (2013), no. 6, 2515–2542. MR 3060893

    2013

  4. [11]

    Shuxing Chen, Stability of transonic shock fronts in two-dimensional Eul er systems , Trans. Amer. Math. Soc. 357 (2005), no. 1, 287–308. MR 2098096

    2005

  5. [12]

    , Transonic shocks in 3-D compressible flow passing a duct with a general section for Euler systems , Trans. Amer. Math. Soc. 360 (2008), no. 10, 5265–5289. MR 2415074

    2008

  6. [13]

    Shuxing Chen and Hairong Yuan, Transonic shocks in compressible flow passing a duct for thre e-dimensional Euler systems , Arch. Ration. Mech. Anal. 187 (2008), no. 3, 523–556. MR 2372813

    2008

  7. [14]

    Courant and K

    R. Courant and K. O. Friedrichs, Supersonic flow and shock waves , Springer-Verlag, New York-Heidelberg, 1976, Reprinting of the 1948 original, Applied Mathematica l Sciences, Vol. 21. MR 0421279

    1976

  8. [15]

    105, Princeton University Press , Princeton, NJ, 1983

    Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlin ear elliptic systems , Annals of Mathematics Studies, vol. 105, Princeton University Press , Princeton, NJ, 1983. MR 717034

    1983

  9. [18]

    Jun Li, Zhouping Xin, and Huicheng Yin, A free boundary value problem for the full Euler system and 2- D transonic shock in a large variable nozzle , Math. Res. Lett. 16 (2009), no. 5, 777–796. MR 2576697

    2009

  10. [19]

    , On transonic shocks in a nozzle with variable end pressures , Comm. Math. Phys. 291 (2009), no. 1, 111–150. MR 2530157

    2009

  11. [20]

    Differential Equations 248 (2010), no

    , On transonic shocks in a conic divergent nozzle with axi-sym metric exit pressures , J. Differential Equations 248 (2010), no. 3, 423–469. MR 2557901 3-D AXISYMMETRIC TRANSONIC SHOCK 71

    2010

  12. [21]

    , Transonic shocks for the full compressible Euler system in a general two-dimensional de Laval nozzle , Arch. Ration. Mech. Anal. 207 (2013), no. 2, 533–581. MR 3005323

    2013

  13. [22]

    Jian-Guo Liu and Wei-Cheng Wang, Characterization and regularity for axisymmetric solenoi dal vector fields with application to Navier-Stokes equation , SIAM J. Math. Anal. 41 (2009), no. 5, 1825–1850. MR 2564196

    2009

  14. [23]

    Li Liu, Gang Xu, and Hairong Yuan, Stability of spherically symmetric subsonic flows and trans onic shocks under multidimensional perturbations , Adv. Math. 291 (2016), 696–757. MR 3459028

    2016

  15. [24]

    Hyperbolic Differ

    Li Liu and Hairong Yuan, Stability of cylindrical transonic shocks for the two-dime nsional steady compressible Euler system , J. Hyperbolic Differ. Equ. 5 (2008), no. 2, 347–379. MR 2420002

    2008

  16. [25]

    Zhouping Xin, Wei Yan, and Huicheng Yin, Transonic shock problem for the Euler system in a nozzle , Arch. Ration. Mech. Anal. 194 (2009), no. 1, 1–47. MR 2533922

    2009

  17. [26]

    Zhouping Xin and Huicheng Yin, Transonic shock in a nozzle. I. Two-dimensional case , Comm. Pure Appl. Math. 58 (2005), no. 8, 999–1050. MR 2143525

    2005

  18. [27]

    , Three-dimensional transonic shocks in a nozzle , Pacific J. Math. 236 (2008), no. 1, 139–193. MR 2398992

    2008

  19. [28]

    Differential Equations 245 (2008), no

    , The transonic shock in a nozzle, 2-D and 3-D complete Euler sy stems, J. Differential Equations 245 (2008), no. 4, 1014–1085. MR 2427405

    2008

  20. [29]

    Hairong Yuan, On transonic shocks in two-dimensional variable-area duct s for steady Euler system , SIAM J. Math. Anal. 38 (2006), no. 4, 1343–1370. MR 2274487

    2006

  21. [30]

    66 (2007), no

    , Transonic shocks for steady Euler flows with cylindrical sym metry, Nonlinear Anal. 66 (2007), no. 8, 1853–1878. MR 2307052

    2007

  22. [31]

    Real World Appl

    , A remark on determination of transonic shocks in divergent n ozzles for steady compressible Euler flows, Nonlinear Anal. Real World Appl. 9 (2008), no. 2, 316–325. MR 2382380

    2008

  23. [32]

    Yong P ark, Department of Mathematics, POSTECH, San 31, Hyojadong, Namgu, Pohang, Gyung- buk, Republic of Korea 37673 Email address : pipablue@postech.ac.kr

    Hairong Yuan and Qin Zhao, Stabilization effect of frictions for transonic shocks in ste ady compressible euler flows passing three-dimensional ducts , (2018). Yong P ark, Department of Mathematics, POSTECH, San 31, Hyojadong, Namgu, Pohang, Gyung- buk, Republic of Korea 37673 E...

    2018

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.