Pith. sign in

REVIEW 1 major objections 4 minor 67 references

Stability of oppositely-propagating pair of Hill's spherical vortices

T0 review · 1 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read This paper proves that a distant, odd-symmetric pair of Hill's spherical vortices remains stable in 3D axisymmetric Euler flow and separates at nearly the single-vortex speed.

desk verdict New result with a real but fixable gap: Theorem 1.1 as stated is not established, only a positive-part version; the proof needs to absorb constants. read the letter →

arxiv 2507.19935 v2 pith:YEHYIZI6 submitted 2025-07-26 math.AP

Young-Jin Sim This is my paper · ORCID
classification math.AP MSC 35Q3176B47
keywords stabilityHill'ssphericalvortexaxisymmetricEulerequationsinteractionenergykinetic-energymaximizeroddsymmetryanti-parallelflowshiftfunctionestimate
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 proves that two Hill's spherical vortices of opposite sign and odd symmetry, initially far apart, keep their individual shapes for all time while moving apart in 3D axisymmetric Euler flow without swirl. The precise statement is a stability theorem with a shift function: closeness to a translated antipodal pair at time zero implies the same closeness, up to a time-dependent translation, at every later time, and an estimate on the shift shows each vortex travels at nearly the single-vortex speed $2/15$ with an error that grows at most like $C\varepsilon(t+1)$. A sympathetic reader should care because it is a genuine multi-vortex stability result in three dimensions: it shows coherent vortex structures can survive while their mutual interaction weakens as they separate, and it identifies the mechanism that enforces the persistence. The proof also gives a sharp estimate of the propagation speed, tying the pair's separation rate to the well-known speed of a single Hill vortex.

What carries the argument

The load-bearing mechanism is the decomposition of the odd-symmetric solution as $\xi=\xi_+-\xi_-$ with $\xi_+$ supported in $z>0$, together with the interaction energy $E_{\mathrm{inter}}(t)=E[\xi_+(t),\xi_-(t)]$. Conservation of total kinetic energy gives $E[\xi(t)]=2E[\xi_+(t)]-2E_{\mathrm{inter}}(t)$; because the impulse $\|r^2\xi_+(t)\|_{L^1}$ decreases monotonically, the energy of each half cannot exceed the maximal value by more than a small amount once the interaction energy is small. The variational theorems imported from the single-vortex theory identify that maximum with Hill's vortex: the scaled Hill vortex is the unique maximizer among axisymmetric vorticities with fixed impulse and circulation bounds, and almost-maximizing sequences are compact up to translation. For the speed estimate, the center of mass $z_c(t)$ satisfies $|\dot z_c(t)-W_H|\lesssim \varepsilon+1/(\tau(t)-1)$, and a bootstrap using the stability theorem and $d\gg\varepsilon^{-1}$ propagates the inequality $|\dot z_c(t)-W_H|<C\varepsilon$ to all times.

What would settle it

Take any fixed $\varepsilon>0$ and consider initial data $\lambda\xi_H(\cdot-d e_z)-\lambda\xi_H(\cdot+d e_z)$ with $d\to\infty$ and $\lambda=1+\varepsilon'$ for small $\varepsilon'$. Theorem 1.1 and 1.2 predict the positive part stays within $\varepsilon$ of a translate of $\xi_H$ in $L^1\cap L^2\cap L^1_w$ for all time and the shift obeys $|\tau(t)-d-\frac{2}{15}t|<C\varepsilon'(t+1)$. A resolved axisymmetric Euler simulation that finds a finite time at which the distance to the family of translates exceeds $\varepsilon$, or a shift error growing faster than the stated linear bound, would falsify the claim; checking several $\lambda$ values would also test the asserted optimality of the $\varepsilon$ exponent.

Watch

Extended reading notes

Core claim

The paper establishes that an odd-symmetric pair of Hill's spherical vortices is stable in the 3D incompressible axisymmetric Euler equations without swirl. Theorem 1.1 states that for every $\varepsilon>0$ there exist $\delta=\delta(\varepsilon)>0$ and $d_0=d_0(\varepsilon)>1$ such that, whenever $d\ge d_0$ and an axisymmetric initial data $\xi_0$ with $\xi_0(r,z)=-\xi_0(r,-z)\ge0$ for $z\ge0$ satisfies $\|\xi_0-(\xi_H(\cdot-d e_z)-\xi_H(\cdot+d e_z))\|_{L^1\cap L^2\cap L^1_w}<\delta$, the solution admits a shift function $\tau$ with $\tau(0)=d$ and $\|\xi(t)-(\xi_H(\cdot-\tau(t)e_z)-\xi_H(\cdot+\tau(t)e_z))\|_{L^1\cap L^2\cap L^1_w}<\varepsilon$ for every $t\ge0$. Theorem 1.2 adds that under an $L^\infty$ bound and a support-localization assumption the shift satisfies $|\tau(t)-\tau(0)-\frac{2}{15}t|<C\varepsilon(t+1)$, and Remark 1.3 argues the exponent of $\varepsilon$ in this estimate is optimal. The proof works by showing the interaction energy between the upper and lower halves remains uniformly small, so each half keeps nearly the maximal kinetic energy, at which point the variational characterization of Hill's vortex as the unique energy maximizer under fixed impulse and circulation forces the profile to stay near a translate.

Load-bearing premise

The proof depends on the theorem, taken from the single-vortex stability theory, that among all axisymmetric vorticity fields with fixed impulse and circulation bounds Hill's vortex is the unique kinetic-energy maximizer and that near-maximizing sequences are compact up to translation; if that characterization failed at the parameter range used, the contradiction argument in Section 3 would place the positive part near a maximizer set rather than near a translated Hill vortex.

Editorial extensions

If this is right

  • Any sufficiently small odd-symmetric perturbation of a distant antipodal Hill pair remains, for all time, within $\varepsilon$ of the same pair with a shifted separation; the pair is Lyapunov stable up to translation.
  • Under the extra $L^\infty$ and support assumptions, the separation shift is linear in time to leading order: $\tau(t)\approx d+\frac{2}{15}t$, with error $C\varepsilon(t+1)$.
  • The linear-in-$\varepsilon$ exponent in the shift bound is optimal, since a $\lambda$-strength Hill pair realizes a matching error of order $|\lambda-1|(t+1)$.
  • The same interaction-energy estimate gives the asymptotic energy identity $E[\xi_+(t)]\to \frac12 E[\xi(0)]$ and convergence of the impulse, which the paper proposes as the starting point for proving long-time approach to a scaled Hill vortex.
  • The method improves the shift-function estimate available for a single Hill vortex from $\varepsilon^{1/2}$ to $\varepsilon$.

Reading between the lines

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

  • A natural extension the author does not pursue is a line of $N$ alternating Hill vortices: non-neighbour interactions decay faster than nearest-neighbour ones, so the same interaction-energy bootstrap might prove stability and linear separation for longer stacks.
  • The proof leaves open the near-odd case where $\xi_0(r,z)$ and $-\xi_0(r,-z)$ are nonnegative but the data are not exactly odd-symmetric; the author expects finite-time stability, but a global statement would need a new mechanism to control impulse growth, since the monotonicity lemma uses exact symmetry.
  • Because the optimality discussion ties the shift error to $\lambda$-strength Hill vortices, a concrete numerical test of the theorem would be to simulate that one-parameter family and measure whether the shift error grows linearly in $|\lambda-1|$; this would test the exponent without needing the full stability statement.
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

1 major / 4 minor

Summary. The paper proves a stability result for a pair of oppositely propagating Hill's spherical vortices in the 3D incompressible axisymmetric Euler equations without swirl. Theorem 1.1 asserts that if an odd-symmetric axisymmetric initial datum is close in L1∩L2∩L1w to the pair ξ_H(·−d e_z)−ξ_H(·+d e_z) for large d, then the unique weak solution remains close to a shifted pair for all time. Theorem 1.2 adds that the shift function propagates at approximately the single-vortex speed W_H=2/15. The proof combines the impulse monotonicity lemma (Lemma 2.1), the interaction-energy identity, and the variational compactness results of [17] that characterize Hill's vortex as the unique energy maximizer. Section 3 contains a contradiction compactness proof; Section 4 derives the shift estimate via a bootstrap on the center of mass.

Significance. If established, the result is a strong contribution to multi-vortex stability in 3D axisymmetric Euler: it is a stability theorem for a separating pair of Hill's vortices and uses a nontrivial interaction-energy mechanism. The paper ships rigorous proofs of the new ingredients (impulse monotonicity, interaction-energy estimates, compactness argument), and the sharpness discussion in Remark 1.3 is plausible. The reliance on the imported variational theorems of [17] is substantial but legitimate. In its current form, however, the paper's main theorem as stated is not fully supported by the proof due to a factor-of-two mismatch between the positive-part estimate and the full-pair distance.

major comments (1)
  1. [Section 3.2, end of proof of Theorem 1.1] The proof establishes only inf_{τ≥1} ∥⟨ξ(t)⟩+−ξ_H(·−τez)∥_{L1∩L2∩L1w} < ε, and then asserts the theorem's conclusion for the full odd-symmetric solution. These are not equivalent. Writing d(t)=⟨ξ(t)⟩+−ξ_H(·−τ(t)ez), odd symmetry gives ∥ξ(t)−(ξ_H(·−τ(t)ez)−ξ_H(·+τ(t)ez))∥_{L1∩L2∩L1w} = 2∥d(t)∥_{L1}+√2∥d(t)∥_{L2}+2∥d(t)∥_{L1w}, which can be as large as twice the positive-part norm. The constants δ(ε) and d0(ε) are not adjusted (for example, by applying Theorem 3.1 with target ε/2), so the stated ε-closeness does not follow from the displayed estimates. This is a statement-proof gap in the central theorem, though it appears repairable by rerunning the scaling argument with a smaller target.
minor comments (4)
  1. [Section 1.1, Theorem 1.1] The proof does not explicitly ensure that the shift function satisfies τ(0)=d; one should set τ(0)=d and note that the initial-data assumption gives the required bound at t=0.
  2. [Section 3.1, Step [3] around (3.7)] The definition of a_n is typeset ambiguously as 'an =∥⟨ξn(0)⟩+−ξµH(·−dnez)∥1/2 2'; it should read a_n = ∥⟨ξn(0)⟩+−ξµH(·−dnez)∥_{L^2}^{1/2} or similar, so that the limit in the preceding display is actually zero.
  3. [Section 4.2, Step 1] The particle-trajectory map φ(t,·) is invoked without proof of existence or uniqueness; for the class of weak solutions at hand this follows from the known log-Lipschitz regularity of the velocity, but a citation or a short justification should be added.
  4. [Section 1.4.1] The text states 'written by M. J. M. Hill in 1984' while reference [42] is dated 1894; the year should be corrected.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the main stability result rests on external variational theorems from [17], not on fitted inputs or self-citations.

full rationale

No circular steps were found. The proof of Theorem 1.1 imports the energy-maximizer uniqueness theorem (Theorem 2.5) and the compactness theorem for maximizing sequences (Theorem 2.8) from Choi's paper [17]; those are rigorous published results by an author not overlapping with the present paper, so this is inheritance of external support rather than self-citation or an ansatz smuggled in via citation. The two-dimensional strategy from [22] is cited only as a structural analogue, and the proof here supplies its own interaction-energy estimates. The shift estimate in Section 4 is self-contained: W_H = 2/15 enters as the known traveling speed of Hill's vortex, not as a fitted parameter, and the center-of-mass evolution is derived from the Biot-Savart law and the smallness assumption (4.3) supplied by Theorem 1.1. The sharpness example in Remark 1.3 legitimately uses the exact family lambda*xi_H as a lower-bound benchmark. The paper does contain minor self-citations to [21] and [23], both co-authored by Sim, but they are used only as pointers to analogous Sadovskii-vortex results and carry no load in the proof, so the score is 2 rather than 0. One separate issue is flagged for completeness but is not circularity: the proof of Theorem 1.1 ends with an estimate on the positive part <xi(t)>_+ only (end of Section 3.2), while the theorem states a bound for the full odd-symmetric pair; by odd symmetry the full L1 and L1_w norms are twice the positive-part norms and the L2 norm is sqrt(2) times, so the stated conclusion does not follow as written from the displayed inequality. That is a correctness gap, not a reduction of the claim to its inputs, and it does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted and no new entities are introduced. The argument imports a cluster of prior rigorous results from [17] and [30], together with the structural template from [22]; these are published results rather than assumptions invented for this paper.

assumptions (6)
  • standard math Existence and uniqueness of axisymmetric weak solutions in L-infinity_t(L1 cap L-infinity) for data in L1 cap L-infinity cap L1_w with r xi_0 in L-infinity
    The solution class in which Theorems 1.1 and 1.2 are stated. Invoked in Section 1.2 via [17, Remark 3.5].
  • standard math Variational energy-maximizer characterization: for mu nu^{-5/3} lambda^{2/3} <= M0, all maximizers in S_{mu,nu,lambda} are z-translations of the scaled Hill vortex
    Imported in Section 2.3 and used in Section 3 step [3] to identify the limit maximizer as xi_H^mu.
  • standard math Compactness of maximizing sequences and energy estimates from [17, Theorems 2.8 and 3.1, Lemma 2.3]
    Used in Section 3 to extract a translated Hill vortex from energy convergence and to transfer L1-cap-L2-cap-L1_w closeness to energy closeness.
  • standard math Uniform velocity bound ||K[xi]||_infinity less than or similar to ||xi||_L-infinity^{1/2} ||xi||_L1^{1/4} ||r^2 xi||_1^{1/4} from Feng and Sverak [30]
    Used in Remark 2.2 and Lemma 4.1 Step 1 to control particle displacement and support growth.
  • domain assumption Odd symmetry of initial data xi_0(r,z) = -xi_0(r,-z) >= 0 for z >= 0
    Defines the anti-parallel flow class; guarantees impulse monotonicity (Lemma 2.1) and positive interaction energy.
  • domain assumption For Theorem 1.2, support condition supp xi_0 subset of {d-2 <= |x3| <= d+2} and ||xi_0||_infinity <= M
    Needed in Lemma 4.1 to bound the initial support spread and the particle displacement term I(t); the bootstrap loop keeps tau(t) > d/2.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Stability of oppositely-propagating pair of Hill's spherical vortices." pith.science (2026). https://pith.science/paper/YEHYIZI6

@misc{pith2026250719935,
  author       = {Pith},
  title        = {Pith review of: Stability of oppositely-propagating pair of Hill's spherical vortices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YEHYIZI6}},
  note         = {Machine review of arXiv:2507.19935}
}
abstract

We establish the stability of a pair of Hill's spherical vortices moving away from each other in 3D incompressible axisymmetric Euler equations without swirl. Each vortex in the pair propagates away from its odd-symmetric counterpart, while keeping its vortex profile close to Hill's vortex. This is achieved by analyzing the evolution of the interaction energy of the pair and combining it with the compactness of energy-maximizing sequences in the variational problem concerning Hill's vortex. The key strategy is to confirm that, if the interaction energy is initially small enough, the kinetic energy of each vortex in the pair remains so close to that of a single Hill's vortex for all time that each vortex profile stays close to the energy maximizer: Hill's vortex. An estimate of the propagating speed of each vortex in the pair is also obtained by tracking the center of mass of each vortex. The estimate can be understood as optimal in the sense that the power exponent of the $\varepsilon$--the small perturbation measured in the ($L^1\cap L^2$+impulse) norm--appearing in the error bound cannot be improved.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

67 extracted references · 59 canonical work pages

  1. [17]

    Stability of Hill’s spherical vortex

    Kyudong Choi. Stability of Hill’s spherical vortex. Comm. Pure Appl. Math., 77(1):52–138, 2024. 24

    work page 2024

  2. [22]

    Stability of vortex quadrupoles with odd-odd symmetry

    Kyudong Choi, In-Jee Jeong, and Yao Yao. Stability of vortex quadrupoles with odd-odd symmetry. arXiv:2409.19822, 2024

    arXiv 2024

  3. [1]

    Stability of Lamb dipoles

    Ken Abe and Kyudong Choi. Stability of Lamb dipoles. Arch. Ration. Mech. Anal., 244(3):877–917, 2022

    work page 2022

  4. [2]

    Stability for multiple Lamb dipoles

    Ken Abe, In-Jee Jeong, and Yao Yao. Stability for multiple Lamb dipoles. arXiv:2507.16474, 2025

    arXiv 2025

  5. [3]

    D. J. Acheson. Instability of vortex leapfrogging. European Journal of Physics, 21(3):269, May 2000

    work page 2000

  6. [4]

    On the existence of leapfrogging pair of circular vortex filaments

    Masashi Aiki. On the existence of leapfrogging pair of circular vortex filaments. Studies in Applied Mathematics, 143(3):213–243, 2019

    work page 2019

  7. [5]

    Amick and L

    Charles J. Amick and L. Edward Fraenkel. The uniqueness of Hill’s spherical vortex. Archive for Rational Mechanics and Analysis, 92:91– 119, 1986

    work page 1986

  8. [6]

    Motion of three vortices

    Hassan Aref. Motion of three vortices. Physics of Fluids, 22(3):393–400, mar 1979

    work page 1979

Show all 67 references
  1. [7]

    Point vortex dynamics: A classical mathematics playground

    Hassan Aref. Point vortex dynamics: A classical mathematics playground. Journal of Mathematical Physics, 48(6):065401, 06 2007

    2007

  2. [8]

    Kadtke, Ireneusz Zawadzki, Laurence J

    Hassan Aref, James B. Kadtke, Ireneusz Zawadzki, Laurence J. Campbell, and Bruno Eckhardt. Point vortex dynamics: recent results and open problems. Fluid Dynamics Research, 3(1):63–74, 1988

    1988

  3. [9]

    Barry, Glen R

    Anna M. Barry, Glen R. Hall, and C. Eugene Wayne. Relative equilibria of the (1+ N)-vortex problem. J. Nonlinear Sci., 22(1):63–83, 2012

    2012

  4. [10]

    Behring and Roy H

    Brandon M. Behring and Roy H. Goodman. Stability of leapfrogging vortex pairs: A semi-analytic approach. Phys. Rev. Fluids, 4:124703, Dec 2019

    2019

  5. [11]

    Leapfrogging vortex rings as scaling limit of Euler equations

    Paolo Butt `a, Guido Cavallaro, and Carlo Marchioro. Leapfrogging vortex rings as scaling limit of Euler equations. SIAM J. Math. Anal., 57(1):789–824, 2025

    2025

  6. [12]

    H. E. Cabral and D. S. Schmidt. Stability of relative equilibria in the problem of N + 1 vortices. SIAM J. Math. Anal., 31(2):231–250, 1999/00

    1999

  7. [13]

    Dynamics and leapfrogging phenomena of multiple helical vortices for 3D incompress- ible Euler equations

    Daomin Cao, Junhong Fan, Guolin Qin, and Jie Wan. Dynamics and leapfrogging phenomena of multiple helical vortices for 3D incompress- ible Euler equations. arXiv:2505.12240, 2025

    2025 arXiv

  8. [14]

    Traveling vortex pairs for 2D incompressible Euler equations

    Daomin Cao, Shanfa Lai, and Weicheng Zhan. Traveling vortex pairs for 2D incompressible Euler equations. Calc. Var.Partial Differential Equations, 60(5):Paper No. 190, 16, 2021

    2021

  9. [15]

    Celli, E

    M. Celli, E. A. Lacomba, and E. P ´erez-Chavela. On polygonal relative equilibria in the N-vortex problem. J. Math. Phys., 52(10):103101, 8, 2011

    2011

  10. [16]

    Cheng, J

    M. Cheng, J. Lou, and T. T. Lim. Numerical simulation of head-on collision of two coaxial vortex rings. Fluid Dyn. Res., 50(6):065513, 24, 2018

    2018

  11. [18]

    On vortex stretching for anti-parallel axisymmetric flows

    Kyudong Choi and In-Jee Jeong. On vortex stretching for anti-parallel axisymmetric flows. Amer. J. Math., to appear (arXiv: 2110.09079), 2021

    2021 arXiv

  12. [19]

    Infinite growth in vorticity gradient of compactly supported planar vorticity near Lamb dipole

    Kyudong Choi and In-Jee Jeong. Infinite growth in vorticity gradient of compactly supported planar vorticity near Lamb dipole. Nonlinear Anal. Real World Appl., 65:Paper No. 103470, 20, 2022

    2022

  13. [20]

    Filamentation near Hill’s vortex

    Kyudong Choi and In-Jee Jeong. Filamentation near Hill’s vortex. Comm. Partial Differential Equations, 48(1):54–85, 2023

    2023

  14. [21]

    On Existence of Sadovskii Vortex Patch: A Touching Pair of Symmetric Counter-Rotating Uniform Vortices

    Kyudong Choi, In-Jee Jeong, and Young-Jin Sim. On Existence of Sadovskii Vortex Patch: A Touching Pair of Symmetric Counter-Rotating Uniform Vortices. Ann. PDE, 11(2):Paper No. 18, 2025

    2025

  15. [23]

    Existence and stability of Sadovskii vortices: from vortex patches to regular vorticity

    Kyudong Choi, Young-Jin Sim, and Kwan Woo. Existence and stability of Sadovskii vortices: from vortex patches to regular vorticity. arXiv:2507.00910, 2025

    2025

  16. [24]

    Head-on collision of two coaxial vortex rings: experiment and computation

    Chin-Chou Chu, Chi-Tzung Wang, Chien-Cheng Chang, Ray-Yu Chang, and Wen-Tyzh Chang. Head-on collision of two coaxial vortex rings: experiment and computation. Journal of Fluid Mechanics, 296:39–71, 1995

    1995

  17. [25]

    Leapfrogging vortex rings for the three-dimensional incompressible Euler equations

    Juan D ´avila, Manuel del Pino, Monica Musso, and Juncheng Wei. Leapfrogging vortex rings for the three-dimensional incompressible Euler equations. Communications on Pure and Applied Mathematics, 77(10):3843–3957, October 2024

    2024

  18. [26]

    Manhar R. Dhanak. Stability of a regular polygon of finite vortices. J. Fluid Mech., 234:297–316, 1992

    1992

  19. [27]

    On the dynamics of leapfrogging vortex rings.arXiv:2503.21604, 2025

    Martin Donati, Lars Eric Hientzsch, Christophe Lacave, and Evelyne Miot. On the dynamics of leapfrogging vortex rings.arXiv:2503.21604, 2025

    2025 arXiv

  20. [28]

    Global in Time Vortex Configurations for the 2D Euler equations

    Juan D ´avila, Manuel del Pino, Monica Musso, and Shrish Parmeshwar. Global in Time Vortex Configurations for the 2D Euler equations. arXiv:2310.07238, 2023

    2023

  21. [29]

    An expanding self-similar vortex configuration for the 2D Euler equations

    Juan D ´avila, Manuel del Pino, Monica Musso, and Shrish Parmeshwar. An expanding self-similar vortex configuration for the 2D Euler equations. arXiv:2410.18220, 2024

    2024 arXiv

  22. [30]

    On the Cauchy problem for axi-symmetric vortex rings

    Hao Feng and Vladim ´ır ˇSver´ak. On the Cauchy problem for axi-symmetric vortex rings. Arch. Ration. Mech. Anal., 215(1):89–123, 2015

    2015

  23. [31]

    L. E. Fraenkel. Examples of steady vortex rings of small cross-section in an ideal fluid. Journal of Fluid Mechanics, 51(1):119–135, 1972

    1972

  24. [32]

    Parker, Andrew W

    Luca Galantucci, Michele Sciacca, Nick G. Parker, Andrew W. Baggaley, and Carlo F. Barenghi. Classical and quantum vortex leapfrogging in two-dimensional channels. Journal of Fluid Mechanics, 912:A9, 2021

    2021

  25. [33]

    V ortex patches choreography for active scalar equations.J

    Claudia Garc ´ıa. V ortex patches choreography for active scalar equations.J. Nonlinear Sci., 31(5):Paper No. 75, 31, 2021

    2021

  26. [34]

    Claudia Garc ´ıa and Susanna V . Haziot. Global bifurcation for corotating and counter-rotating vortex pairs.Comm. Math. Phys., 402(2):1167– 1204, 2023

    2023

  27. [35]

    Co-rotating vortices with N fold symmetry for the inviscid surface quasi- geostrophic equation

    Ludovic Godard-Cadillac, Philippe Gravejat, and Didier Smets. Co-rotating vortices with N fold symmetry for the inviscid surface quasi- geostrophic equation. Indiana Univ. Math. J., 72(2):603–650, 2023

    2023

  28. [36]

    Numerical simulations of two coaxial vortex rings head-on collision

    Hui Guan, Zhi-Jun Wei, Elizabeth Rumenova Rasolkova, and Chui-Jie Wu. Numerical simulations of two coaxial vortex rings head-on collision. Adv. Appl. Math. Mech., 8(4):616–647, 2016

    2016

  29. [37]

    Steady asymmetric vortex pairs for Euler equations

    Zineb Hassainia and Taoufik Hmidi. Steady asymmetric vortex pairs for Euler equations. Discrete Contin. Dyn. Syst., 41(4):1939–1969, 2021

    1939

  30. [38]

    Rigorous derivation of the leapfrogging motion for planar Euler equations

    Zineb Hassainia, Taoufik Hmidi, and Nader Masmoudi. Rigorous derivation of the leapfrogging motion for planar Euler equations. arXiv: 2311.15765, 2023

    2023 arXiv

  31. [39]

    Zineb Hassainia and Miles H. Wheeler. Multipole Vortex Patch Equilibria for Active scalar equations. SIAM Journal on Mathematical Analysis, 54(6):6054–6095, 2022

    2022

  32. [40]

    T. H. Havelock. LII. The stability of motion of rectilinear vortices in ring formation. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 11(70):617–633, 1931

    1931

  33. [41]

    Helmholtz

    H. Helmholtz. ¨Uber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. J. Reine Angew. Math., 55:25–55, 1858

  34. [42]

    M. J. M. Hill. On a Spherical Vortex. Philosophical Transactions of the Royal Society of London Series A, 185:213–245, 1894

  35. [43]

    Existence of corotating and counter-rotating vortex pairs for active scalar equations

    Taoufik Hmidi and Joan Mateu. Existence of corotating and counter-rotating vortex pairs for active scalar equations. Comm. Math. Phys., 350(2):699–747, 2017

    2017

  36. [44]

    Sideris, and Pascal Gamblin

    Dragos ¸ Iftimie, Thomas C. Sideris, and Pascal Gamblin. On the evolution of compactly supported planar vorticity.Comm. Partial Differential Equations, 24(9-10):1709–1730, 1999

    1999

  37. [45]

    V orlesungen¨uber mathematische Physik, volume 1

    Gustav Kirchho ff. V orlesungen¨uber mathematische Physik, volume 1. 1883

  38. [46]

    L. G. Kurakin. Influence of annular boundaries on Thomson’s vortex polygon stability. Chaos, 24(2):023105, 12, 2014

    2014

  39. [47]

    L. G. Kurakin and I. V . Ostrovskaya. Stability of the Thomson vortex polygon with evenly many vortices outside a circular domain.Siberian Mathematical Journal, 51(3):463–474, May 2010

    2010

  40. [48]

    L. G. Kurakin and I. V . Ostrovskaya. On the stability of the system of Thomson’s vortex n-gon and a moving circular cylinder. Russ. J. Nonlinear Dyn., 18(5):905–916, 2022

    2022

  41. [49]

    L. G. Kurakin and V . I. Yudovich. The stability of stationary rotation of a regular vortex polygon.Chaos, 12(3):574–595, 2002

    2002

  42. [50]

    Hydrodynamics

    Horace Lamb. Hydrodynamics. Cambridge Mathematical Library. Cambridge University Press, Cambridge, sixth edition, 1993. With a foreword by R. A. Caflisch [Russel E. Caflisch]

    1993

  43. [51]

    Leoncini, L

    X. Leoncini, L. Kuznetsov, and G. M. Zaslavsky. Motion of three vortices near collapse. Phys. Fluids, 12(8):1911–1927, 2000

    1911

  44. [52]

    C. C. Lin. On the motion of vortices in two dimensions. I. Existence of the Kirchho ff-Routh function. Proc. Nat. Acad. Sci. U.S.A., 27:570– 575, 1941

    1941

  45. [53]

    C. C. Lin. On the motion of vortices in two dimensions. II. Some further investigations on the Kirchho ff-Routh function. Proc. Nat. Acad. Sci. U.S.A., 27:575–577, 1941

    1941

  46. [54]

    Paul K. Newton. The N-vortex problem, volume 145 of Applied Mathematical Sciences. Springer-Verlag, New York, 2001. Analytical techniques. 25

    2001

  47. [55]

    J. Norbury. A steady vortex ring close to Hill’s spherical vortex. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 72, pages 253–284. Cambridge University Press, 1972

    1972

  48. [56]

    J. Norbury. A family of steady vortex rings. Journal of Fluid Mechanics, 57(3):417–431, 1973

    1973

  49. [57]

    E. A. Novikov and Iu. B. Sedov. V ortex collapse.Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, 77:588–597, August 1979

    1979

  50. [58]

    Head-on Collision of Two Vortex Rings

    Yuko Oshima. Head-on Collision of Two Vortex Rings. Journal of the Physical Society of Japan, 44(1):328–331, 1978

    1978

  51. [59]

    Pozrikidis

    C. Pozrikidis. The nonlinear instability of Hill’s vortex. Journal of Fluid Mechanics, 168:337–367, 1986

    1986

  52. [60]

    Linear stability of Hill’s vortex to axisymmetric perturbations

    Bartosz Protas and Alan Elcrat. Linear stability of Hill’s vortex to axisymmetric perturbations. Journal of Fluid Mechanics, 799:579–602, 2016

    2016

  53. [61]

    V ortex rings

    Karim Shari ff and Anthony Leonard. V ortex rings. InAnnual review of fluid mechanics, Vol. 24, pages 235–279. Annual Reviews, Palo Alto, CA, 1992

    1992

  54. [62]

    Zabusky, and Joel H

    Karim Shari ff, Anthony Leonard, Norman J. Zabusky, and Joel H. Ferziger. Acoustics and dynamics of coaxial interacting vortex rings.Fluid Dynamics Research, 3(1):337–343, 1988

    1988

  55. [63]

    Instability of vortex pair leapfrogging

    Laust Tophøj and Hassan Aref. Instability of vortex pair leapfrogging. Physics of Fluids, 25(1):014107, 01 2013

    2013

  56. [64]

    On steady vortex flow in two dimensions

    Bruce Turkington. On steady vortex flow in two dimensions. I, II. Comm. Partial Differential Equations, 8(9):999–1030, 1031–1071, 1983

    1983

  57. [65]

    Variational principles for Hill’s spherical vortex and nearly spherical vortices

    Yieh Hei Wan. Variational principles for Hill’s spherical vortex and nearly spherical vortices. Transactions of the American Mathematical Society, 308(1):299–312, 1988

    1988

  58. [66]

    V ortex motion of the Euler and lake equations.J

    Cheng Yang. V ortex motion of the Euler and lake equations.J. Nonlinear Sci., 31(3):Paper No. 48, 21, 2021

    2021

  59. [67]

    From point vortices to vortex patches in self-similar expanding configurations

    Samuel Zbarsky. From point vortices to vortex patches in self-similar expanding configurations. Comm. Math. Phys., 388(2):707–733, 2021. Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, 50 UNIST-gil, Eony ang-eup, Ulju-gun, Ulsan 44919, ...

    2021

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.