Pith. sign in

REVIEW 2 major objections 4 minor 54 references

Finite-time self-similar implosion of hollow vortices

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read This paper proves that generic collapsing configurations of point vortices desingularize into families of hollow vortices that implode self-similarly in finite time, with their cores shrinking to a point and the internal pressure diverging.

desk verdict First rigorous construction of finite-time hollow-vortex implosion, with a strong self-contained rotating-vortex half and a desingularization theorem whose proof has a load-bearing gap at the Fredholm-index comparison. read the letter →

arxiv 2506.04093 v1 pith:USKJ3PDD submitted 2025-06-04 math.AP

Robin Ming Chen , Samuel Walsh , Miles H. Wheeler This is my paper
classification math.AP MSC 35Q3176B4735B3235R35
keywords finite-timeblowuphollowvorticesself-similarimplosionpointvortexcollapsedesingularizationfreeboundaryEulerequationsbifurcationtheorydynamics
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

Finite-time collapse has long been known for idealized point vortices, but classical Euler evolution of vortex patches is globally regular, so a more physical intermediate model is needed to realize blowup. This paper uses hollow vortices: gas-filled cores surrounded by vortex sheets, where the ideal gas law makes the core pressure diverge as the area shrinks. The authors prove that any non-degenerate collapsing point-vortex configuration can be desingularized: for each small conformal radius there is a genuine hollow-vortex solution, real-analytic in that radius, which inherits the self-similar spiraling implosion of the point vortices. They also prove the existence of two new branches of near-circular m-fold symmetric rotating hollow vortices for every m at least 2, and a rigidity result saying purely circular imploding vortices are locally unique among collapsing vortices with uniform velocity at infinity.

What carries the argument

The machinery is a conformal-mapping reformulation of the self-similar hollow-vortex problem on a fixed circular domain. The unknown vortex boundaries and potentials are encoded through layer-potential densities (μ, ν) on the unit circle, with conformal radius ρ, so the nonlinear free-boundary system becomes an analytic operator equation F(u; ρ) = 0: at ρ = 0 this equation recovers the Kirchhoff–Helmholtz point-vortex algebraic system V(Λ) = 0, and for ρ > 0 it encodes the full hollow-vortex equations. The argument then hinges on a Fredholm-index comparison: at a point-vortex configuration, the linearization of the hollow-vortex operator is claimed to have kernel and cokernel dimensions equal to those of D_λ V(Λ0), so at non-degenerate configurations the implicit function theorem yields the desingularizing family.

What would settle it

Compute dim ker and codim rng of the linearized operator D_u F(u0, 0) from (4.20) at a non-degenerate collapsing point-vortex configuration such as the trio (4.5) or quartet (4.6); any kernel or cokernel dimension beyond dim ker D_λ V(Λ0) would falsify Theorem 1.3 as stated.

Watch

Extended reading notes

Core claim

The central discovery is that finite-time singularity formation, previously confined to idealized point-vortex dynamics, is realized by solutions of the free-boundary hollow-vortex Euler equations. The main theorem constructs, from any non-degenerate configuration of M collapsing point vortices, a real-analytic one-parameter family of hollow vortex configurations (U^ρ, V^ρ, λ^ρ) for ρ in [0, ρ1) that bifurcates from the point-vortex configuration at ρ = 0 and exhibits self-similar finite-time implosion: the vortices spiral toward the origin, their cores shrink to points, and the core pressure diverges as the collapse time approaches. The paper also establishes that for every m ≥ 2 there are two real-analytic branches of near-circular m-fold symmetric rotating hollow vortices, bifurcating from the circular vortex at angular velocities Ω0,± = γ/(2π)(1 ± 1/√m), and that purely circular imploding vortices are locally unique among collapsing vortices with uniform velocity at infinity.

Load-bearing premise

The load-bearing premise is the claim, not proved in this text, that the linearized hollow-vortex operator at a point-vortex configuration has kernel and cokernel dimensions exactly matching the linearized collapsing point-vortex system, via a comparison with the authors' earlier work [9].

Editorial extensions

If this is right

  • For any non-degenerate collapsing point-vortex configuration, in particular the Gröbli trio and the explicit quartet, there exist nearby genuine hollow-vortex solutions with self-similar finite-time implosion and divergent pressure.
  • Since the system is time-reversible, each imploding family also yields expanding hollow-vortex configurations that grow out of a point.
  • Near-circular m-fold rotating hollow vortices exist on two distinct branches for every m ≥ 2; the '+' branch has no critical layers, the '−' branch exhibits cat's-eye streamlines, and on the '−' branch the bifurcation changes from supercritical to subcritical as m grows.
  • The purely circular imploding hollow vortex is locally the only collapsing hollow vortex with uniform velocity at infinity within the m-fold symmetric class.
  • The desingularization gives another rigorous justification that point-vortex dynamics is the effective limit of concentrated vortex motion: the hollow-vortex solutions converge to the point-vortex configuration as the core radius tends to zero.

Reading between the lines

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

  • This suggests a two-tier regularization hierarchy: point vortices collapse in finite time, hollow vortices collapse too, while classical vortex patches remain globally regular; probing whether the bubble pressure law admits nearby polytropic variants might reveal which thermodynamic modeling assumptions the blowup depends on.
  • Because the bifurcation parameter ρ measures the core size, the family K can be read as a measure of how much physical regularization still permits the singular point-vortex collapse; a natural next step the paper does not take is to test whether these constructed hollow vortices are dynamically stable under the Euler evolution.
  • The asymptotic expansion (4.19) gives explicit leading-order formulas for the boundary shapes and circulation densities, so direct numerical simulation of the free-boundary Euler equations for the trio and quartet could test whether the predicted implosion persists beyond infinitesimal core radius.
  • If the rigidity of circular imploding vortices extends beyond m-fold symmetry, it would sharply constrain possible self-similar finite-time singularities in this model.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies finite-time self-similar implosion and steady rotation of hollow vortices in the two-dimensional Euler equations. In Section 3 the authors give an explicit linear analysis of circular hollow vortices: a closed-form dispersion relation, a complete description of kernel and range of the linearized operator, and a Crandall–Rabinowitz bifurcation argument yielding two families of near-circular m-fold rotating hollow vortices for every m ≥ 2. They also prove a local rigidity theorem for circular imploding vortices. In Section 4 they set up a layer-potential formulation of the multi-vortex problem, formulate the collapsing point-vortex system as a finite-dimensional algebraic system, and state a desingularization theorem (Theorem 1.3, precise version Theorem 4.4) asserting that non-degenerate collapsing point-vortex configurations yield real-analytic families of imploding hollow vortices. The paper concludes with two concrete examples, an imploding trio and quartet, obtained by computer-algebra verification of non-degeneracy.

Significance. If the desingularization theorem is correct, this is the first rigorous construction of collapsing hollow vortices and provides new finite-time singularity solutions of the free-boundary Euler equations; it also gives a natural bridge between the classical Gr"obli collapsing point-vortex configurations and the full Euler model. The rotating-sector analysis in Section 3 is especially valuable: the dispersion relation, the kernel/range characterization, and the transversality condition are computed in closed form, and the two-branch structure with its critical-layer behaviour is a genuinely new qualitative feature compared with H-states and V-states. The main caveat is that the central desingularization result rests on an asserted comparison of Fredholm indices with the authors' prior preprint [9]; that comparison is not proved here and is not a formal consequence of the displayed formulas. The paper is nonetheless well written, and the gap appears to be fillable within the manuscript's scope.

major comments (2)
  1. [§4.3, Eq. (4.21)] The proof of Theorem 4.4 hinges on the identities dim ker D_u F(u0,0) = dim ker D_λ V(Λ0) and codim rng D_u F(u0,0) = codim rng D_λ V(Λ0), which are asserted by 'arguing directly as in the proof of [9, Theorem 6.1]'. This comparison is not demonstrated in the text, and it is not a routine consequence of the displayed formulas (4.20): the collapsing problem has a different V, hence different A^0_{kλ} and B^0_{kλ}, and Remark 4.5 explicitly notes that Qρ is O(ρ) rather than O(ρ^2), so the finite-dimensional reduction genuinely differs from the steady case treated in [9]. Since the implicit function theorem is invoked only after (4.21), this is the load-bearing step of Theorem 1.3. The authors should either state and prove the relevant linearized Fredholm comparison in the collapsing setting, or give a detailed reduction to [9, Theorem 6.1] that accounts for the change in V and in the scaling of Qρ, and they should verify the kernel/cokernel dimensions directly in the examples of Corollary 4.6.
  2. [§4.2, after (4.18)] The proof of Theorem 4.4 asserts that univalence of the conformal mapping f on Dρ 'will follow for the small hollow vortex solutions by construction, as f will be near identity', but no argument is supplied. Since Theorem 4.4 claims actual hollow vortex configurations, not merely solutions to the abstract equation F(u,ρ)=0, injectivity of f for small ρ is part of the conclusion. This is likely routine (for example, by combining the non-vanishing of 1+ρZ^ρ_k μ'_k on the boundary with a quantitative near-identity estimate on the circular domain), but as written it is a gap in the passage from F^{-1}(0) to a physical fluid configuration.
minor comments (4)
  1. [Eq. (3.8)] In the displayed definition of F2 in (3.8), the formula appears to be missing the division by |1+Cµ'|^2 that is required by the dynamic condition (2.6c). Please check whether this is a typesetting omission and correct the displayed equation.
  2. [Corollary 4.6] The statement 'Fix ℓ ≥ 1 and α ∈ (0, α)' contains a typo; the second α should be a fixed exponent in (0,1), presumably the same α used in Theorem 3.3 or a new symbol.
  3. [§4.3] The word 'real-anayltic' in the proof of Theorem 4.4 should be 'real-analytic'.
  4. [Appendix A] The higher-order asymptotic formulas (A.2) are described as obtainable by 'an elementary but lengthy calculation', but no derivation is given. These expansions are not needed for the main existence theorems, so this is not a blocking issue, but the paper should state explicitly whether they are rigorously proved or only formally derived.

Circularity Check

1 steps flagged · score 4.0 of 10

The desingularization theorem's load-bearing Fredholm-index comparison (4.21) is asserted by self-citation to the authors' own [9] rather than proved for the collapsing case.

  1. self citation load bearing [Section 4.3, proof of Theorem 4.4, around (4.20)-(4.21)]
    "These agree exactly with the linearized steady hollow vortex system computed in [9, Section 6]; the collapsing case differs only in the form of V and hence A0kλ and B0kλ. Thanks to this similarity, arguing directly as in the proof of [9, Theorem 6.1], we conclude that DuF(u0,0) : X → Y is Fredholm with dim kerDuF(u0,0) = dim kerDλV(Λ0) codim rng DuF(u0,0) = codim rng DλV(Λ0). (4.21)"

    The implicit-function-theorem step that produces the family K requires DuF(u0,0) to be an isomorphism, and the only argument supplied for the dimension identities (4.21) is an appeal to the authors' own preprint [9, Theorem 6.1]. The displayed linearizations (4.20) and Remark 4.5 show that the collapsing problem differs from the steady problem in V, in A0kλ, B0kλ, and in the O(ρ) size of Qρ; the index comparison is therefore not a formal consequence of the formulas written here. Since the theorem's conclusion is obtained by applying the implicit function theorem only after (4.21), the desingularization claim rests on a load-bearing self-citation rather than on a derivation contained in this paper.

full rationale

Section 3 is self-contained: the linearized operator is computed explicitly, the dispersion relation (3.13) is derived, and the Crandall-Rabinowitz transversality check is performed in full; Theorems 1.1 and 1.2 do not reduce to fits or to the target conclusions. The point-vortex collapse inputs in Theorem 1.3 are classical, and the layer-potential formulation (4.7)-(4.17) is explicitly constructed; no fitted parameter is renamed as a prediction. The single genuinely load-bearing gap is the identity (4.21): the proof of Theorem 4.4 needs the kernel and cokernel dimensions of DuF to match those of DλV so that DuF is an isomorphism, but the paper obtains this by 'arguing directly as in the proof of [9, Theorem 6.1]', a self-citation to the authors' own preprint, and Remark 4.5 shows the collapsing problem is not identical to the steady problem in [9]. Because [9] is an external prior work rather than an assumption of the theorem, this is not a case of the conclusion being assumed; however, the central existence result currently rests on an index comparison that is not verified in this text, and Corollary 4.6 verifies only the point-vortex Jacobian, not the hollow-vortex linearized operator. The appropriate circularity score is therefore 4: some load-bearing self-citation, but the central claim retains substantial independent mathematical content and is not forced by construction.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no fitted constants or invented physical entities. Its free parameters are the physical parameters of the model (circulations, angular velocity, collapse time, conformal radius). The main assumptions are the physical model itself, the self-similar ansatz, standard functional-analytic theorems, the cited Fredholm-index comparison from the authors' prior preprint, and computational verification of non-degeneracy of examples.

assumptions (7)
  • domain assumption Hollow vortex model (1.1) with isobaric ideal gas pressure law
    The mathematical model studied in the paper: incompressible irrotational outer flow, pressure constant on each core, and p|V| constant.
  • domain assumption Self-similar ansatz and non-inertial frame (1.2)
    The paper restricts to solutions that become steady in a rotating and contracting frame; this is the class in which all theorems are stated.
  • standard math Crandall-Rabinowitz bifurcation theorem (Theorem 2.1)
    Used in Theorem 3.3 to obtain the two branches of rotating hollow vortices.
  • standard math Layer potential representation with Sokhotski-Plemelj and Privalov theorems (Section 2.3)
    Used to reformulate the free boundary problem as a finite-dimensional bifurcation problem.
  • standard math Near-circular conformal mapping estimates of Gaier/Marchenko and Schauder theory (Lemma 3.6)
    Used to upgrade near-circularity of the domain to C^{1+alpha} control of the conformal map, needed for the rigidity theorem.
  • domain assumption Fredholm index comparison from authors' prior work [9, Theorem 6.1]
    The proof of Theorem 4.4 asserts the kernel/cokernel dimension match for the collapsing case 'arguing directly as in the proof of [9, Theorem 6.1]', a paper by the same authors that is not reproduced here.
  • domain assumption Non-degeneracy of explicit point vortex examples verified by computer algebra
    The examples in Corollary 4.6 require Jacobian computations reported as numerical checks with no code shipped; the quartet determinant is given only approximately (det D_lambda V approx 1.397).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Finite-time self-similar implosion of hollow vortices." pith.science (2026). https://pith.science/paper/USKJ3PDD

@misc{pith2026250604093,
  author       = {Pith},
  title        = {Pith review of: Finite-time self-similar implosion of hollow vortices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/USKJ3PDD}},
  note         = {Machine review of arXiv:2506.04093}
}
abstract

In this paper, we consider the finite-time blowup of hollow vortices. These are solutions of the two-dimensional Euler equations for which the fluid domain is the complement of finitely many Jordan curves $\Gamma_1, \ldots, \Gamma_M$, and such that the flow is irrotational and incompressible, but with a nonzero circulation around each boundary component. The region bounded by $\Gamma_k$ is a ``vortex core'', modeled as a bubble of ideal gas: the pressure is constant in space and inversely proportional to the area of the vortex. This can be thought of as the isobaric approximation assuming isothermal flow. Our results come in two parts. There exist explicit families of purely circular rotating and imploding hollow vortices. Implosion means more precisely that the vortex core shrinks to the origin in finite time, while the absolute value of the pressure simultaneously diverges to infinity. We prove that for any $m \geq 2$, there exist near-circular $m$-fold symmetric rotating hollow vortices. By contrast, for all $m \geq 2$, the purely circular imploding vortices are locally unique among all collapsing vortices with uniform velocity at infinity. The second part concerns configurations of multiple hollow vortices. The existence of configurations of point vortices that collapse into a common point in finite time is classical. We prove that generically, these can be desingularized to yield families of hollow vortex configurations exhibiting self-similar finite-time implosion. Specific examples of an imploding trio and quartet of hollow vortices are given.

Figures

Figures reproduced from arXiv: 2506.04093 by the authors.

Figure 1
Figure 1. Sketched streamline patterns of rotating hollow vortices with m-fold symmetry for m = 2, 3, 4, 5; the first row correspond to solutions on C m,+ rot , while the second row are from C m,− rot . The vortex is the shaded region, and the shape of the vortex and the streamlines are based on the leading-order approximation in (A.2). Note that along the “+” branch, there are no critical layers, whereas on the “−” branch, o… view at source ↗
Figure 2
Figure 2. Sketched fluid particle trajectories, in the spiraling (ξ, T) frame (1.2), for the collapsing hollow vortex trio (a) and quartet (b) in Corollary 4.6. The velocity field U, which has constant divergence 1/κ > 0, is based on the corresponding point vortex configurations (4.5) and (4.6). The Kirchhoff–Helmholtz model for a collapsing configuration reduces to a system of 2M (real) algebraic equations; written abstractl… view at source ↗
Figure 2
Figure 2. 1.3. Plan of the article. The steady fluid domain D is unknown a priori, and so to facilitate the analysis we make a further change of variables in Section 2, recasting the problem on a fixed domain. This comes at the usual expense of complicating the equations. In particular, we take D (complexified in the natural way) to be the image of a conformal mapping f defined on the “circular domain” Dρ := C \ B|ρ| (ζ1) ∪ ·… view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

54 extracted references · 52 canonical work pages

  1. [9]

    R. M. Chen, S. W alsh, and M. H. Wheeler, Desingularization and global continuation for hollow vortices , (2023). Preprint

    work page 2023

  2. [1]

    Aref, Motion of three vortices , The Physics of Fluids, 22 (1979), pp

    H. Aref, Motion of three vortices , The Physics of Fluids, 22 (1979), pp. 393–400

    work page 1979

  3. [2]

    , Point vortex dynamics: a classical mathematics playground , J. Math. Phys., 48 (2007), pp. 065401, 23

    work page 2007

  4. [3]

    , Self-similar motion of three point vortices , Physics of fluids, 22 (2010)

    work page 2010

  5. [4]

    H. Aref, P. K. Newton, M. A. Stremler, T. Tokieda, and D. L. V ainchtein , Vortex Crystals , Adv. Appl. Mech., 39 (2003), pp. 1–79

    work page 2003

  6. [5]

    H. Aref, N. Rott, and H. Thomann , Gr¨ obli’s solution of the three-vortex problem, in Annual review of fluid mechanics, Vol. 24, Annual Reviews, Palo Alto, CA, 1992, pp. 1–20

    work page 1992

  7. [6]

    G. R. Baker, P. G. Saffman, and J. S. Sheffield , Structure of a linear array of hollow vortices of finite cross-section, Journal of Fluid Mechanics, 74 (1976), pp. 469–476

    work page 1976

  8. [7]

    T. B. Benjamin , Hamiltonian theory for motions of bubbles in an infinite liquid , J. Fluid Mech., 181 (1987), pp. 349–379

    work page 1987

Show all 54 references
  1. [8]

    Buffoni and J

    B. Buffoni and J. Toland , Analytic theory of global bifurcation , Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2003. An introduction

    2003

  2. [10]

    Cie´slak, P

    T. Cie´slak, P. Kokocki, and W. S. O ˙za´nski, Well-posedness of logarithmic spiral vortex sheets , J. Differential Equations, 389 (2024), pp. 508–539

    2024

  3. [11]

    Constantin and W

    A. Constantin and W. A. Strauss , Periodic traveling gravity water waves with discontinuous vorticity , Arch. Rational Mech. Anal., 202 (2011), pp. 133–175

    2011

  4. [12]

    M. G. Crandall and P. H. Rabinowitz , Bifurcation from simple eigenvalues , J. Func. Anal., 8 (1971), pp. 321–340

    1971

  5. [13]

    D. G. Crowdy and C. C. Green , Analytical solutions for von k´ arm´ an streets of hollow vortices, Physics of Fluids, 23 (2011), p. 126602

    2011

  6. [14]

    D. G. Crowdy, S. G. Llewellyn Smith, and D. V. Freilich , Translating hollow vortex pairs , Eur. J. Mech. B Fluids, 37 (2013), pp. 180–186

    2013

  7. [15]

    D. G. Crowdy, R. B. Nelson, and V. S. Krishnamurthy , ‘H-states’: exact solutions for a rotating hollow vortex, J. Fluid Mech., 913 (2021), pp. Paper No. R5, 11

    2021

  8. [16]

    Delort, Existence de nappes de tourbillon en dimension deux , J

    J.-M. Delort, Existence de nappes de tourbillon en dimension deux , J. Amer. Math. Soc., 4 (1991), pp. 553–586

    1991

  9. [17]

    D´avila, M

    J. D´avila, M. del Pino, M. Musso, and S. Parmeshwar , An expanding self-similar vortex configuration for the 2d Euler equations , (2024). IMPLODING HOLLOW VORTICES 29

    2024

  10. [18]

    Elling, Algebraic spiral solutions of 2d incompressible Euler , J

    V. Elling, Algebraic spiral solutions of 2d incompressible Euler , J. Differential Equations, 255 (2013), pp. 3749– 3787

    2013

  11. [19]

    , Self-similar 2d Euler solutions with mixed-sign vorticity , Comm. Math. Phys., 348 (2016), pp. 27–68

    2016

  12. [20]

    Gaier, On conformal mapping of nearly circular regions , Pacific J

    D. Gaier, On conformal mapping of nearly circular regions , Pacific J. Math., 12 (1962), pp. 149–162

    1962

  13. [21]

    Garc´ıa and S

    C. Garc´ıa and S. V. Haziot , Global bifurcation for corotating and counter-rotating vortex pairs , Comm. Math. Phys., 402 (2023), pp. 1167–1204

    2023

  14. [22]

    Specielle Probleme ¨ uber die Bewegung geradliniger paralleler Wirbelf¨ aden

    R. H. Goodman , An English translation of Gr¨ obli’s Ph.D. dissertation: “Specielle Probleme ¨ uber die Bewegung geradliniger paralleler Wirbelf¨ aden ”, 2024

    2024

  15. [23]

    Gr ¨obli, Specielle Probleme ¨ uber die Bewegung geradliniger paralleler Wirbelf¨ aden, vol

    W. Gr ¨obli, Specielle Probleme ¨ uber die Bewegung geradliniger paralleler Wirbelf¨ aden, vol. 8, Druck von Z¨ urcher und Furrer, 1877

  16. [24]

    Hassainia, N

    Z. Hassainia, N. Masmoudi, and M. H. Wheeler , Global bifurcation of rotating vortex patches , Comm. Pure Appl. Math., 73 (2020), pp. 1933–1980

    2020

  17. [25]

    Helmholtz, ¨Uber integrale der hydrodynamischen gleichungen, welche den wirbelbewegungen entsprechen

    H. Helmholtz, ¨Uber integrale der hydrodynamischen gleichungen, welche den wirbelbewegungen entsprechen. , Journal f¨ ur die reine und angewandte Mathematik, 55 (1858), pp. 25–55

  18. [26]

    Hern´andez-Gardu˜no and E

    A. Hern´andez-Gardu˜no and E. A. Lacomba , Collisions and regularization for the 3-vortex problem , Journal of Mathematical Fluid Mechanics, 9 (2007), pp. 75–86

    2007

  19. [27]

    H ¨older, ¨Uber die unbeschr¨ ankte fortsetzbarkeit einer stetigen ebenen bewegung in einer unbegrenzten inkom- pressiblen fl¨ ussigkeit, Mathematische Zeitschrift, 37 (1933), pp

    E. H ¨older, ¨Uber die unbeschr¨ ankte fortsetzbarkeit einer stetigen ebenen bewegung in einer unbegrenzten inkom- pressiblen fl¨ ussigkeit, Mathematische Zeitschrift, 37 (1933), pp. 727–738

    1933

  20. [28]

    Jeong and A

    I.-J. Jeong and A. R. Said , Logarithmic spirals in 2d perfect fluids , arXiv preprint arXiv:2302.09447, (2023)

    2023

  21. [29]

    S. S. Kallyadan and P. Shukla , Self-similar vortex configurations: Collapse, expansion, and rigid-vortex motion, Phys. Rev. Fluids, 7 (2022), p. 114701

    2022

  22. [30]

    Kimura, Similarity solution of two-dimensional point vortices , Journal of the Physical Society of Japan, 56 (1987), pp

    Y. Kimura, Similarity solution of two-dimensional point vortices , Journal of the Physical Society of Japan, 56 (1987), pp. 2024–2030

    1987

  23. [31]

    V. S. Krishnamurthy and M. A. Stremler , Finite-time collapse of three point vortices in the plane , Regul. Chaotic Dyn., 23 (2018), pp. 530–550

    2018

  24. [32]

    Kudela, Collapse of n-point vortices in self-similar motion , Fluid Dyn

    H. Kudela, Collapse of n-point vortices in self-similar motion , Fluid Dyn. Res., 46 (2014), pp. 031414, 16

    2014

  25. [33]

    Lai and M

    C.-C. Lai and M. I. Weinstein , Asymmetric deformations of a perturbed spherical bubble in an incompressible fluid, arXiv preprint arXiv:2408.03787, (2024)

    2024 arXiv

  26. [34]

    L. Lichtenstein , ¨Uber einige existenzprobleme der hydrodynamik homogener, unzusammendr¨ uckbarer, rei- bungsloser fl¨ ussigkeiten und die helmholtzschen wirbels¨ atze, Mathematische Zeitschrift, 23 (1925), pp. 89–154

    1925

  27. [35]

    S. G. Llewellyn Smith and D. G. Crowdy , Structure and stability of hollow vortex equilibria , J. Fluid Mech., 691 (2012), pp. 178–200

    2012

  28. [36]

    Marchenko, Sur la representation conforme , CR Acad

    A. Marchenko, Sur la representation conforme , CR Acad. Sci. USSR, 1 (1935), pp. 289–290

    1935

  29. [37]

    R. B. Nelson, V. S. Krishnamurthy, and D. G. Crowdy , The corotating hollow vortex pair: steady merger and break-up via a topological singularity , J. Fluid Mech., 907 (2021), pp. Paper No. A10, 21

    2021

  30. [38]

    P. K. Newton , The N -vortex problem: Analytical techniques , vol. 145 of Applied Mathematical Sciences, Springer-Verlag, New York, 2001

    2001

  31. [39]

    Res., 46 (2014), pp

    , Point vortex dynamics in the post-Aref era , Fluid Dyn. Res., 46 (2014), pp. 031401, 11

    2014

  32. [40]

    Novikov and I

    E. Novikov and I. B. Sedov , Vortex collapse, Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, 77 (1979), pp. 588–597

    1979

  33. [41]

    E. A. Overman, II , Steady-state solutions of the Euler equations in two dimensions. II. Local analysis of limiting V -states, SIAM J. Appl. Math., 46 (1986), pp. 765–800

    1986

  34. [42]

    Park, Quantitative estimates for uniformly-rotating vortex patches , Adv

    J. Park, Quantitative estimates for uniformly-rotating vortex patches , Adv. Math., 411 (2022), pp. Paper No. 108779, 55

    2022

  35. [43]

    H. C. Pocklington , The configuration of a pair of equal and opposite hollow straight vortices of finite cross section moving steadily through fluid , Proc. Cambridge Philos. Soc., 8 (1894), pp. 178–187

  36. [44]

    Prosperetti, The thermal behaviour of oscillating gas bubbles , J

    A. Prosperetti, The thermal behaviour of oscillating gas bubbles , J. Fluid Mech., 222 (1991), pp. 587–616

    1991

  37. [45]

    P. G. Saffman , Vortex dynamics, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992

    1992

  38. [46]

    F. Shao, D. Wei, and Z. Zhang , Self-similar algebraic spiral solution of 2-d incompressible euler equations , arXiv preprint arXiv:2305.05182, (2023)

    2023 arXiv

  39. [47]

    , Self-similar algebraic spiral vortex sheets of 2-d incompressible euler equations , arXiv preprint arXiv:2505.03309, (2025)

    2025 arXiv

  40. [48]

    C. H. Su , Motion of fluid with constant vorticity in a singly-connected region , The Physics of Fluids, 22 (1979), pp. 2032–2033

    1979

  41. [49]

    Traizet, Hollow vortices and minimal surfaces , J

    M. Traizet, Hollow vortices and minimal surfaces , J. Math. Phys., 56 (2015), pp. 083101, 18

    2015

  42. [50]

    W ang, G

    Y. W ang, G. Zhang, and M. Zhou, Boundary regularity of uniformly rotating vortex patches and an unstable elliptic free boundary problem , arXiv, 2306.03498 (2024). 30 R. M. CHEN, S. W ALSH, AND M. H. WHEELER

    2024 arXiv

  43. [51]

    W. Wolibner, Un theor` eme sur l’existence du mouvement plan d’un fluide parfait, homog` ene, incompressible, pendant un temps infiniment long , Mathematische Zeitschrift, 37 (1933), pp. 698–726

    1933

  44. [52]

    H. M. Wu, E. A. Overman II, and N. J. Zabusky , Steady-state solutions of the Euler equations in two dimensions: rotating and translating V -states with limiting cases. I. Numerical algorithms and results , J. Comput. Phys., 53 (1984), pp. 42–71

    1984

  45. [53]

    V. I. Yudovich , Non-stationary flows of an ideal incompressible fluid , Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 3 (1963), pp. 1032–1066

    1963

  46. [54]

    Zannetti, M

    L. Zannetti, M. Ferlauto, and S. G. Llewellyn Smith , Hollow vortices in shear , J. Fluid Mech., 809 (2016), pp. 705–715. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260 Email address : mingchen@pitt.edu Department of Mathematics, University of Missou...

    2016

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.