Pith. sign in

REVIEW 2 major objections 6 minor 1 cited by

Remarks on radial symmetry of stationary and uniformly-rotating solutions for the 2D Euler equation

T0 review · 2 major / 6 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 A uniformly rotating solution of the 2D Euler equation with compactly supported vorticity must be radially symmetric whenever its angular velocity lies at or outside half the range of the vorticity, with stationary solutions radial up to…

desk verdict A real extension of the Duke 2021 rigidity theorem—sign-changing vorticity and irregular patches—with a sound central proof; the gaps are omitted details, not errors. read the letter →

arxiv 2506.05034 v1 pith:SSGB3JQH submitted 2025-06-05 math.AP

Boquan Fan , Yuchen Wang , Weicheng Zhan This is my paper
classification math.AP MSC 35Q3135B0676B47
keywords radialsymmetry2DEulerequationsvortexpatchuniformlyrotatingsolutionscontinuousSteinersymmetrizationlocalsign-changingvorticitysharpthreshold
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 a rigidity theorem for two-dimensional ideal fluids: if a compactly supported vorticity distribution rotates uniformly (or is stationary) and its angular velocity satisfies $\Omega \le \inf \omega_0/2$ or $\Omega \ge \sup \omega_0/2$, then the vorticity must be radially symmetric, up to translation when $\Omega = 0$. The result covers vortex patches whose boundaries are merely finite unions of disjoint Jordan curves, so the boundaries may be fractal or otherwise rough, and it covers $C^2$ smooth compactly supported vorticity, allowing vorticity that changes sign. This matters because it identifies a precise region in which non-radial steady or uniformly rotating structures of the planar Euler equation can exist: outside that angular-velocity range, the vorticity is forced to be round regardless of patch irregularity. The paper also constructs non-radial sign-changing patches by bifurcation, showing that the thresholds in the theorem cannot be widened.

What carries the argument

The load-bearing object is the continuous Steiner symmetrization (CStS), a one-parameter family of rearrangements $T_t$ that continuously shrinks each vertical slice of a set or function toward its midpoint until the classical Steiner symmetrization is reached at $t = \infty$. The paper uses it through a local-symmetry criterion: if a compactly supported function in $H^1$ has vanishing first-order derivative of its Dirichlet energy under the CStS, then the function is locally symmetric. The proof verifies that derivative is zero for the truncated stream function $(u-c_0)_+$ by a truncation argument and an estimate, imported from earlier work, that the symmetric difference $L^2(U^t_k \Delta U_k)$ between a far-away superlevel set and its symmetrization is bounded by a constant times $H^1(\partial U_k)$ times $R_k t$; the far-away level sets are nearly circular because $u$ behaves like $\Omega|x|^2 + O(|x|^{-1})$ at infinity. Local symmetry, together with the structure theorem saying a locally symmetric function consists of countable annuli on which it is radially decreasing, is upgraded to global radiality by a weak-superharmonicity and maximum-principle argument. An appendix applies an abstract bifurcation-from-simple-eigenvalue theorem to the contour equations for two nested patches and exhibits non-radial branches at angular velocities approaching the thresholds, proving sharpness.

What would settle it

Compute $L^2(U^t \Delta U)$ for a Koch-snowflake-shaped superlevel set of a uniformly rotating patch with $\Omega$ outside $[\inf \omega_0/2, \sup \omega_0/2]$ and check whether it obeys the bound $\le C H^1(\partial U) R t$ used in the proof; finding a counterexample to that displacement estimate, or any non-radial patch satisfying the contour equation (1.3) in that parameter range, would refute the rigidity theorem.

Watch

Extended reading notes

Core claim

The central claim is that radial symmetry is forced by the superharmonic or subharmonic nature of the stream function $u = N * \omega_0 + \frac{\Omega}{2}|x|^2$, not by positivity of vorticity or by smoothness of the patch boundary. Under the hypothesis $\Omega \le \inf \omega_0/2$ the function $u$ is weakly superharmonic on the plane, and under $\Omega \ge \sup \omega_0/2$ the function $-u$ is; the integral equations (1.6) and (1.7) make $u$ constant on each patch boundary or on each regular level set of $\omega_0$. The paper proves that such a $u$ is locally symmetric in every direction by running the continuous Steiner symmetrization and showing the Dirichlet energy is unchanged to first order, so a local-symmetry criterion applies. Once local symmetry is combined with a maximum-principle rigidity argument for weakly superharmonic functions, $u$ must be radial, and hence $\omega_0$ is radial, up to translation when $\Omega = 0$. This covers multi-patch solutions with sign-changing weights and boundaries that are finite unions of mutually disjoint Jordan curves, as well as $C^2$ smooth compactly supported vorticity, thereby extending previous rigidity results beyond the regime of nonpositive angular velocities and nonnegative vorticity.

Load-bearing premise

The whole argument hinges on a bound, imported from earlier work, that each far-away level set of the stream function is displaced by the symmetrization procedure at speed at most its own radius; if that displacement bound fails for sets with merely Jordan boundaries, the proof that the energy stays unchanged to leading order collapses.

Editorial extensions

If this is right

  • Any stationary multi-patch solution with the integral-equation structure (1.6), arbitrary signs, and Jordan-curve boundaries must be radially symmetric up to translation, so lopsided stationary multi-component patches cannot exist in that class.
  • Any uniformly rotating smooth solution with compact support and $\Omega$ outside $[\inf \omega_0/2, \sup \omega_0/2]$ is radial, removing the earlier requirements of nonnegative vorticity and nonpositive angular velocity.
  • The thresholds in the patch theorem are sharp: the bifurcation construction in Appendix A yields non-radial sign-changing double patches at angular velocities approaching the boundary values, so the interval is exactly the window in which non-radial uniform rotation is possible.
  • Because radiality is deduced from the stream function rather than from a global vorticity-strength relation $\omega = f(\psi)$, the method avoids the need for a globally defined profile function linking vorticity to the stream function.

Reading between the lines

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

  • A testable extension: if the level-set displacement estimate behind the CStS proof extends to other interaction kernels, the same recipe should force radial symmetry for uniformly rotating patches of related active scalar models whenever the corresponding stream function is super- or subharmonic; the Euler-specific input is mainly the far-field expansion of the stream function.
  • The bifurcation formulas in the appendix suggest that for sign-changing double patches the non-radial families are discrete and indexed by the symmetry mode, with angular velocities accumulating at the threshold values as the mode number grows; one could test this prediction by computing higher-mode branches.
  • It is plausible that the $C^2$ condition in the smooth theorem is used only to ensure level sets are regular, so the smooth result may persist for weaker regularity whenever the level-set equation (1.7) is imposed directly as the definition of a uniformly rotating solution.
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 / 6 minor

Summary. The paper proves rigidity results for uniformly rotating solutions of the 2D incompressible Euler equation with compactly supported vorticity. Theorems 1.1 and 1.4 assert that any multi-patch solution satisfying (1.6) or any C^2 smooth solution satisfying (1.7) must be radially symmetric, up to translation when the angular velocity is zero, whenever Omega <= inf omega_0 / 2 or Omega >= sup omega_0 / 2. The proof represents the stream function as u = N * omega_0 + (Omega/2)|x|^2 and uses Brock's continuous Steiner symmetrization together with a local-symmetry criterion to show that u is radial. The superharmonic or subharmonic nature of u, forced by the angular-velocity condition, allows truncations at far-away level sets; estimates on the slow motion of level sets under symmetrization show that the Dirichlet-energy derivative vanishes, yielding local symmetry, and a maximum-principle lemma upgrades this to global radial symmetry. An appendix sketches a bifurcation result indicating that the angular-velocity bounds are sharp.

Significance. If correct, the paper substantially extends the rigidity theorems of Gomez-Serrano, Park, Shi, and Yao ([20]): it removes the non-negativity restriction on vorticity and allows patch boundaries that are merely finite unions of Jordan curves, including irregular boundaries. The continuous Steiner symmetrization method is conceptually clean and is applied here to the unbounded plane by a truncation argument, which is a genuine technical contribution. The paper also supplies a bifurcation construction that plausibly shows sharpness of the Omega-interval. The main ideas and the overall proof architecture are sound and original. However, the manuscript as written has gaps in the proof of the subharmonic case and in a key technical lemma used for irregular patch boundaries; these need to be repaired before the theorems as stated are fully established.

major comments (2)
  1. [Section 3.2 (and 4.2)] The proof in the subharmonic case begins with the statement 'In this case, Omega >= 0'. This does not follow from the condition Omega >= sup omega_0 / 2 when the vorticity is sign-definite negative: if omega_0 <= 0 and not identically zero, then sup omega_0 < 0 and Omega may be negative. In that situation u = N * omega_0 + (Omega/2)|x|^2, while subharmonic, still tends to -infinity, but the reduction to \tilde u = -u gives a function tending to +infinity, and the previous proof relies on the stream function tending to -infinity to define the far-away level sets Gamma_c and the compactly supported truncations (u - c0)_+. Consequently the proof as written covers only Omega >= 0, whereas Theorems 1.1 and 1.4 do not assume sign-changing vorticity. Please either restrict the theorems to sign-changing vorticity, matching the abstract, or supply the missing argument for sign-definite negative vorticity, for example by time-reversal symmetry.
  2. [Lemma 2.12] The proof of Lemma 2.12 asserts that '(u^t - gamma_k)_+ 1_{U_i} is a rearrangement of (u - gamma_k)_+ 1_{U_i} for all sufficiently small t' by the definition of the CStS. This does not follow from the stated conditions on U_i. The continuous Steiner symmetrization of a product of a function with the indicator of a set is not in general the product of the symmetrized function with the same indicator, and for an arbitrary domain U_i satisfying G_i subset U_i subset V and {u > gamma_k} intersect U_i = G_i, the symmetrized level sets may leave U_i. The identities (2.11)-(2.14), and hence the o(t) estimate (2.6), rely on this assertion, so the proof of the lemma is incomplete. This lemma is used at (3.14) and (4.7), so the gap affects the treatment of irregular patch boundaries. The lemma can be repaired by choosing each U_i to be a neighbourhood of the closure of G_i that is separated from the other level sets; with that choice (u - gamma_k)_+ vanishes outside U_i and, for small t, so does its continuous Steiner symmetrization, making the rearrangement identity true. Please state and justify this choice explicitly.
minor comments (6)
  1. [Section 4.1] The sentence 'In this case, Omega >= 0' should read Omega <= 0: compact support of omega_0 gives inf omega_0 <= 0, and the estimates that follow use only |Omega|, so this is a typo rather than a mathematical issue.
  2. [After (3.10)] The key estimate L^2(U^t_k Delta U_k) <= 2 H^1(Gamma_{gamma_k}) R_k t is imported from Lemma 4.1 and Eq. (4.2) of [20] without proof. Since it is a load-bearing quantitative input for (3.11) and the analogous estimates in Section 3.1, please state it as a lemma and include a proof sketch, for example from the interval property in Definition 2.2(iv) together with the C^1 nature of the far level sets.
  3. [Appendix A] Several computations in the bifurcation analysis are only said to follow from [12] (for instance the smoothness of the nonlinear map F and the derivation of the linearized operator), and there are typos such as 'lineazied', 'eignvalues', and the duplicated phrase 'at the angular velocity at the angular velocity'. Please expand the derivations or state explicitly which results of [12] are being invoked.
  4. [Equation (3.5)] The notation [U]^t appears in the display but is not defined and should be U^t.
  5. [Section 3.1, Case 1] In the case Omega = 0, the proof sets A = (1/2 pi) int omega_0 and assumes A is nonzero without comment. If omega_0 is identically zero the result is trivial; otherwise the condition Omega <= inf omega_0 / 2 with Omega = 0 forces omega_0 >= 0, so A > 0. Please add this one-line justification.
  6. [Section 4.1, step-function approximation] The approximation omega_0 by step functions w_n = sum_j alpha_j 1_{D_j} should allow the coefficients alpha_j to depend on n, and the approximation properties (a)-(d) should be stated with a clear link to the regular-level-set decomposition of omega_0, since the current description is terse.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main rigidity theorem is derived from external symmetrization and elliptic tools, with no self-citation carrying a load-bearing step.

full rationale

I traced the derivation chain from the integral conditions (1.6) and (1.7) through Sections 3 and 4. The central claim is that, under the angular-velocity bounds Omega <= inf omega0/2 or Omega >= sup omega0/2, the stream function u = N*omega0 + (Omega/2)|x|^2 is radial. These bounds are not fitted or inferred from the conclusion; they are used to make u weakly superharmonic or weakly subharmonic through -Delta u = omega0 - 2Omega. The local-symmetry machinery comes from Brock's continuous Steiner symmetrization, cited as external work in Propositions 2.5, 2.7, and 2.9. The delicate level-set motion estimate L2(U_k^t Delta U_k) <= C H^1(dU_k) R_k t is imported from [20, Lemma 4.1 and Eq. (4.2)], an external source whose content is an estimate about continuous symmetrization, not the radial-symmetry theorem being proved. The only self-citation is [16], mentioned in the introduction as a prior use of the same methodological idea; it is not used as a theorem on which the proof depends. The phrase 'Omega >= 0' at the start of Section 4.1 appears to be a sign typo for the compact-support case and is not used in a load-bearing way. The bifurcation appendix uses the standard Crandall-Rabinowitz theorem rather than importing any uniqueness result from the authors' own prior work. No equation is defined in terms of the target result, and no fitted parameter is renamed as a prediction. I therefore find no circular step requiring a quote-and-reduction analysis.

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

No free parameters are fitted to data; all constants are explicit or universal. The argument imports the continuous Steiner symmetrization toolbox from Brock and the truncation/approximation estimates from [20] and [16]. These are external results rather than outputs of this paper, so the axiom ledger reflects reliance on established literature, not ad hoc postulates.

assumptions (5)
  • standard math The continuous Steiner symmetrization family T_t exists on R and R^2 and satisfies the properties in Proposition 2.5: equimeasurability, monotonicity, semigroup, interval property, L^p continuity, and Dirichlet-energy monotonicity.
    Used throughout Sections 3 and 4 as the foundation of the method; these properties are proved in the cited works of Brock, not reproved here.
  • standard math Brock's local symmetry criterion: if the CStS Dirichlet-energy derivative vanishes at t=0, then the function is locally symmetric, and locally symmetric functions decompose into radially symmetric annuli (Propositions 2.7 and 2.9).
    Invoked in Section 3 to pass from energy monotonicity to local symmetry of the stream function.
  • standard math Weakly superharmonic functions that are C^1 and piecewise smooth obey the maximum principle and Hopf-type boundary estimates needed in Lemma 2.8 and Lemma 2.12.
    Used to conclude that local symmetry plus superharmonicity forces global radial symmetry on disks and to estimate boundary-layer integrals.
  • standard math For the smooth case, every C^2 compactly supported vorticity can be approximated by step functions whose level-set boundaries are regular level sets of the vorticity, with the approximation properties listed on page 2997 of [20].
    Section 4.1 explicitly adopts this approximation from [20] to reduce the smooth case to patch-type domains.
  • domain assumption In the Omega = 0 case of Section 3.1, the total circulation integral of w0 is nonzero whenever w0 is nonnegative and not identically zero; this makes the leading term in the far-field expansion of grad u nondegenerate.
    The proof of the level-set estimates for Omega = 0 requires A = (1/2 pi) integral w0 to be nonzero; this follows from compact support and w0 >= 0, but the paper could state it more explicitly.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Remarks on radial symmetry of stationary and uniformly-rotating solutions for the 2D Euler equation." pith.science (2026). https://pith.science/paper/SSGB3JQH

@misc{pith2026250605034,
  author       = {Pith},
  title        = {Pith review of: Remarks on radial symmetry of stationary and uniformly-rotating solutions for the 2D Euler equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SSGB3JQH}},
  note         = {Machine review of arXiv:2506.05034}
}
abstract

We prove that any uniformly rotating solution of the 2D incompressible Euler equation with compactly supported vorticity $\omega$ must be radially symmetric whenever its angular velocity satisfies $\Omega \in (-\infty,\inf \omega / 2] \cup \, [ \sup \omega / 2, +\infty )$, in both the patch and smooth settings. This result extends the rigidity theorems established in \cite{Gom2021MR4312192} (\textit{Duke Math. J.},170(13):2957-3038, 2021), which were confined to the case of non-positive angular velocities and non-negative vorticity. Moreover, our results do not impose any regularity conditions on the patch beyond requiring that its boundary consists of Jordan curves, thereby refining the previous result to encompass irregular vortex patches.

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On stationary Quasi-Geostrophic Shallow-Water flows

    math.AP 2026-07 accept novelty 7.0 of 10

    Non-trivial m-fold doubly-connected stationary vortex patches are proven to exist for the quasi-geostrophic shallow-water equations via bifurcation from annuli.

Reference graph

Works this paper leans on

40 extracted references · 35 canonical work pages · cited by 1 Pith paper

  1. [12]

    de la Hoz, T

    F. de la Hoz, T. Hmidi, J. Mateu, and J. Verdera. Doubly connected V -states for the planar Euler equations. SIAM J. Math. Anal. , 48(3):1892–1928, 2016

    work page 1928

  2. [20]

    G´ omez-Serrano, J

    J. G´ omez-Serrano, J. Park, J. Shi, and Y. Yao. Symmetry in stationary and uniformly rotating solutions of active scalar equations. Duke Math. J. , 170(13):2957–3038, 2021

    work page 2021

  3. [19]

    G´ omez-Serrano, J

    J. G´ omez-Serrano, J. Park, and J. Shi. Existence of non-trivial non-concentrated compactly supported stationary solutions of the 2d euler equation with finite energy. Mem. Amer. Math. Soc. , in press

    work page

  4. [1]

    Bedrossian and V

    J. Bedrossian and V. Vicol. The mathematical analysis of the incompressible Euler and Navier-Stokes equations—an introduction, volume 225 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2022. 2022

    work page 2022

  5. [2]

    A. L. Bertozzi and P. Constantin. Global regularity for vortex patches.Comm. Math. Phys., 152(1):19– 28, 1993

    work page 1993

  6. [3]

    Brock and P

    F. Brock and P. Tak´ aˇ c. Symmetry and stability of non-negative solutions to degenerate elliptic equa- tions in a ball. Proc. Amer. Math. Soc. , 150(4):1559–1575, 2022

    work page 2022

  7. [4]

    F. Brock. Continuous Steiner-symmetrization. Math. Nachr. , 172:25–48, 1995

    work page 1995

  8. [5]

    F. Brock. Continuous rearrangement and symmetry of solutions of elliptic problems. Proc. Indian Acad. Sci. Math. Sci. , 110(2):157–204, 2000

    work page 2000

Show all 40 references
  1. [6]

    F. Brock. Rearrangements and applications to symmetry problems in PDE. In Handbook of differ- ential equations: stationary partial differential equations. Vol. IV , Handb. Differ. Equ., pages 1–60. Elsevier/North-Holland, Amsterdam, 2007

    2007

  2. [7]

    F. Brock. Symmetry for a general class of overdetermined elliptic problems. NoDEA Nonlinear Dif- ferential Equations Appl. , 23(3):Art. 36, 16, 2016

    2016

  3. [8]

    Buffoni and J

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

    2003

  4. [9]

    J-Y. Chemin. Persistance de structures g´ eom´ etriques dans les fluides incompressibles bidimensionnels. Ann. Sci. ´Ecole Norm. Sup. (4) , 26(4):517–542, 1993

    1993

  5. [10]

    Choffrut and L

    A. Choffrut and L. Sz´ ekelyhidi, Jr. Weak solutions to the stationary incompressible Euler equations. SIAM J. Math. Anal. , 46(6):4060–4074, 2014

    2014

  6. [11]

    Crandall and P.H

    M.G. Crandall and P.H. Rabinowitz. Bifurcation from simple eigenvalues. J. Functional Analysis , 8:321–340, 1971

    1971

  7. [13]

    De Lellis and L

    C. De Lellis and L. Jr. Sz´ ekelyhidi, The Euler equations as a differential inclusion.Ann. of Math. (2) , 170(3):1417–1436, 2009

    2009

  8. [14]

    De Lellis and L

    C. De Lellis and L. Jr. Sz´ ekelyhidi. Theh-principle and the equations of fluid dynamics. Bull. Amer. Math. Soc. (N.S.) , 49(3):347–375, 2012

    2012

  9. [15]

    De Lellis and L

    C. De Lellis and L. Jr. Sz´ ekelyhidi. Dissipative continuous Euler flows.Invent. Math., 193(2):377–407, 2013

    2013

  10. [16]

    B. Fan, Y. Wang, and W. Zhan. Radial symmetry of stationary and uniformly-rotating solutions to the 2D Euler equation in a disc. arXiv:2412.05973

    arXiv

  11. [17]

    Fraenkel

    L.E. Fraenkel. An introduction to maximum principles and symmetry in elliptic problems, volume 128 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 2000

    2000

  12. [18]

    Gidas, W

    B. Gidas, W. Ni, and L. Nirenberg. Symmetry and related properties via the maximum principle. Comm. Math. Phys. , 68(3):209–243, 1979

    1979

  13. [21]

    C. Gui, C. Xie and H. Xu. On a classification of steady solutions to two-dimensional Euler equations. arXiv:2405.15327

    arXiv

  14. [22]

    Hamel and N

    F. Hamel and N. Nadirashvili. A Liouville theorem for the Euler equations in the plane. Arch. Ration. Mech. Anal., 233(2):599–642, 2019

    2019

  15. [23]

    Hamel and N

    F. Hamel and N. Nadirashvili. Circular flows for the Euler equations in two-dimensional annular domains, and related free boundary problems. J. Eur. Math. Soc. , 25(1):323–368, 2023

    2023

  16. [24]

    T. Hmidi. On the trivial solutions for the rotating patch model. J. Evol. Equ. , 15(4):801–816, 2015. 30 BOQUAN F AN, YUCHEN W ANG, WEICHENG ZHAN

    2015

  17. [25]

    Hmidi and J

    T. Hmidi and J. Mateu. Degenerate bifurcation of the rotating patches. Adv. Math. , 302:799–850, 2016

    2016

  18. [26]

    Hmidi, J

    T. Hmidi, J. Mateu, and J. Verdera. Boundary regularity of rotating vortex patches. Arch. Ration. Mech. Anal., 209(1):171–208, 2013

    2013

  19. [27]

    B. Kawohl. Rearrangements and convexity of level sets in PDE, volume 1150 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1985

    1985

  20. [28]

    Krantz and H.R

    S.G. Krantz and H.R. Parks. The geometry of domains in space. Birkh¨ auser Advanced Texts: Basler Lehrb¨ ucher. Birkh¨ auser Boston, Inc., Boston, MA, 1999

    1999

  21. [29]

    Lieb and M

    E.H. Lieb and M. Loss. Analysis, volume 14 of Graduate Studies in Mathematics . American Mathe- matical Society, Providence, RI, Second Edition, 2001

    2001

  22. [30]

    Majda and A.L

    A.J. Majda and A.L. Bertozzi. Vorticity and incompressible flow, volume 27 of Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002

    2002

  23. [31]

    Marchioro and M

    C. Marchioro and M. Pulvirenti. Mathematical theory of incompressible nonviscous fluids, volume 96 of Applied Mathematical Sciences. Springer-Verlag, New York, 1994

    1994

  24. [32]

    Pommerenke

    C. Pommerenke. Boundary behaviour of conformal maps, volume 299 of Grundlehren der mathema- tischen Wissenschaften. Springer-Verlag, Berlin, 1992

    1992

  25. [33]

    D. Ruiz. Symmetry results for compactly supported steady solutions of the 2D Euler equations. Arch. Ration. Mech. Anal., 247(3):Paper No. 40, 25, 2023

    2023

  26. [34]

    H. Sagan. Space-filling curves. Universitext. Springer-Verlag, New York, 1994

    1994

  27. [35]

    J. Serra. Radial symmetry of solutions to diffusion equations with discontinuous nonlinearities. J. Differential Equations, 254(4):1893–1902, 2013

    1902

  28. [36]

    A.Y. Solynin. Exercises on the theme of continuous symmetrization. Comput. Methods Funct. Theory, 20(3-4):465–509, 2020

    2020

  29. [37]

    Y. Wang, X. Xu, and M. Zhou. Degenerate bifurcations of two-fold doubly-connected uniformly rotating vortex patches. arXiv:2212.01869

    arXiv

  30. [38]

    Wang and W

    Y. Wang and W. Zhan. On the rigidity of the 2D incompressible Euler equations. arXiv:2307.00197v2

    arXiv

  31. [39]

    Wolibner

    W. Wolibner. Un theor` eme sur l’existence du mouvement plan d’un fluide parfait, homog` ene, incom- pressible, pendant un temps infiniment long. Mathematische Zeitschrift , 37(1):698–726, 1933

    1933

  32. [40]

    Yudovich

    V.I. Yudovich. Non-stationary flows of an ideal incompressible fluid. ˇZ. Vyˇ cisl. Mat i Mat. Fiz. , 3:1032–1066, 1963. School of Mathematical Sciences and Institute of Natural Sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, P. R. China Email address : ...

    1963

Pith tools

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