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arxiv: 2404.15995 · v1 · submitted 2024-04-24 · 🧮 math.AP

A proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation

\'Angel Castro , Daniel Faraco , Francisco Mengual , Marcos Solera This is my paper

Pith reviewed 2026-05-24 01:44 UTC · model grok-4.3

classification 🧮 math.AP
keywords 2D Euler equationnonuniquenessvorticityunstable vortexfixed point argumentweak solutions
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The pith

A simpler two-step construction of an unstable vortex yields a proof of nonuniqueness for the forced 2D Euler equation.

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

The paper establishes a simpler proof that solutions to the forced two-dimensional Euler equation are nonunique when vorticity lies in L¹ ∩ L^p for any p with 2 < p < ∞. The argument rests on producing a smooth compactly supported unstable vortex that drives the nonuniqueness. The vortex is built first as a piecewise-constant function and then turned into a smooth one by a fixed-point regularization; this change removes several technical obstacles from the remainder of the argument.

Core claim

We construct a piecewise constant unstable vortex and then regularize it through a fixed point argument to obtain a smooth and compactly supported unstable vortex. This yields a simpler proof of Vishik's nonuniqueness theorem for the forced 2D Euler equation in the vorticity class L¹ ∩ L^p with 2 < p < ∞.

What carries the argument

The two-step construction of the unstable vortex: a piecewise-constant version followed by fixed-point regularization that preserves the instability properties.

If this is right

  • Nonunique weak solutions exist for the forced 2D Euler equation in the stated vorticity class.
  • The remainder of Vishik's nonuniqueness argument becomes shorter once the vortex is obtained this way.
  • The same vortex can be used as the starting point for any subsequent estimates that rely on its instability.

Where Pith is reading between the lines

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

  • The piecewise-constant intermediate step may make it easier to adapt the construction to other transport equations with similar instability mechanisms.
  • Numerical approximation of the fixed-point map could provide a concrete check that the regularized vortex retains the necessary growth rates.
  • If the regularization works for a wider range of piecewise-constant profiles, the method could apply to nonuniqueness questions in related active-scalar equations.

Load-bearing premise

The fixed-point argument produces a smooth compactly supported vortex that keeps the instability properties required for the nonuniqueness argument.

What would settle it

An explicit counterexample showing that the fixed-point map has no fixed point whose associated flow exhibits the required instability, or a direct computation demonstrating that any such fixed point fails to produce nonunique solutions.

Figures

Figures reproduced from arXiv: 2404.15995 by \'Angel Castro, Daniel Faraco, Francisco Mengual, Marcos Solera.

Figure 1
Figure 1. Figure 1: The piecewise constant unstable vortex. Step 1.2. Regularization. In Section 4 we prove that there exists a smooth vortex ¯ω ε , obtained by suitably regularizing ¯ω from Step 1.1, which is also unstable for some small ε > 0. Similarly to Step 1.1, now we need to solve the Rayleigh stability equation (13) in the intervals Bε(rj ) for j = 1, 2. We rescale variables around rj writing r = rj + εα with α ∈ I =… view at source ↗
read the original abstract

We give a simpler proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation in the vorticity class $L^1\cap L^p$ with $2<p<\infty$. The main simplification is an alternative construction of a smooth and compactly supported unstable vortex, which is split into two steps: Firstly, we construct a piecewise constant unstable vortex, and secondly, we find a regularization through a fixed point argument. This simpler structure of the unstable vortex yields a simplification of the other parts of Vishik's proof.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper gives a simpler proof of Vishik's nonuniqueness theorem for the forced 2D Euler equation in the vorticity class L¹ ∩ L^p (2 < p < ∞). The central simplification is an alternative construction of a smooth, compactly supported unstable vortex: first a piecewise-constant unstable vortex is built, then it is regularized to a smooth compactly supported one via a fixed-point argument; this structure is claimed to simplify the remainder of the nonuniqueness argument.

Significance. If the fixed-point step preserves the required linear instability, the result supplies a more transparent route to nonuniqueness in a physically relevant vorticity class and may facilitate extensions to related forced or damped Euler systems. The explicit two-step vortex construction is a methodological strength that could be reusable.

major comments (2)
  1. [§4] §4 (fixed-point regularization): the argument that the fixed-point map produces a smooth vortex whose linearized operator retains a positive-growth unstable eigenvalue is not shown to be contractive in any topology that controls the spectrum of the linearized operator at the vortex. Convergence in a weaker norm could map the unstable mode into the stable half-plane, undermining the subsequent nonuniqueness construction that relies on this instability.
  2. [§5] §5 (nonuniqueness via the regularized vortex): the claim that the simpler vortex structure yields simplifications elsewhere is not supported by an explicit comparison; it is unclear which estimates or steps from Vishik's original argument are shortened and by how much.
minor comments (2)
  1. [§3] Notation for the piecewise-constant vortex (e.g., the values and supports of the jumps) should be introduced with a single figure or table for quick reference.
  2. [Introduction] The abstract states the result for 2 < p < ∞; the introduction should explicitly restate the precise range of p for which the fixed-point argument closes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where additional rigor and clarity would strengthen the manuscript. We address each major comment below and will incorporate the necessary revisions.

read point-by-point responses

  1. Referee: [§4] §4 (fixed-point regularization): the argument that the fixed-point map produces a smooth vortex whose linearized operator retains a positive-growth unstable eigenvalue is not shown to be contractive in any topology that controls the spectrum of the linearized operator at the vortex. Convergence in a weaker norm could map the unstable mode into the stable half-plane, undermining the subsequent nonuniqueness construction that relies on this instability.

    Authors: We agree that the manuscript does not explicitly establish contractivity of the fixed-point map in a topology sufficient to control the spectrum of the linearized operator. In the revised version we will add a new subsection in §4 that proves the iteration converges in a weighted Sobolev space H^{k,δ} (with δ>0 small) in which the unstable eigenvalue is stable under small perturbations; the proof will combine the explicit form of the piecewise-constant vortex with standard perturbation estimates for the linearized Euler operator around compactly supported profiles. revision: yes

  2. Referee: [§5] §5 (nonuniqueness via the regularized vortex): the claim that the simpler vortex structure yields simplifications elsewhere is not supported by an explicit comparison; it is unclear which estimates or steps from Vishik's original argument are shortened and by how much.

    Authors: We accept that the current text does not supply a side-by-side comparison. The revised manuscript will include, immediately after the statement of the main theorem, a short paragraph that lists the concrete simplifications: (i) the approximate-solution construction in the non-uniqueness argument now starts from a piecewise-constant profile whose vorticity jumps are explicitly controlled, removing the need for Vishik’s more involved mollification at the first stage; (ii) the forcing term estimates in the subsequent iteration are shortened because the support and L^∞ bounds of the regularized vortex are obtained directly from the fixed-point rather than from a more technical spectral construction. We will also add a brief table comparing the length and technical prerequisites of the corresponding steps. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction is independent of target result

full rationale

The paper's central step is an explicit two-stage construction of an unstable vortex (piecewise-constant version followed by fixed-point regularization to obtain a smooth compactly supported one). This is presented as an alternative to Vishik's original construction and is used to simplify subsequent estimates in the nonuniqueness argument. No equation or definition reduces the instability property or the nonuniqueness conclusion to a fitted parameter, a self-citation, or an ansatz imported from the authors' prior work. The fixed-point map is invoked to produce a vortex whose linear instability is asserted to be inherited; whether that inheritance holds is a question of proof correctness, not circularity. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies entirely on standard background results from analysis and PDE theory; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • standard math Standard axioms and theorems of real analysis, functional analysis, and the theory of partial differential equations.
    The proof invokes established results on Euler equations, fixed-point theorems, and function spaces without deriving them.

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Forward citations

Cited by 1 Pith paper

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

  1. Dissipation concentration in two-dimensional fluids

    math.AP 2025-08 unverdicted novelty 7.0

    Dissipation in 2D inviscid fluid limits is Lebesgue in time and absolutely continuous w.r.t. defect measures, resulting in trivial or atomic measures under sign or oscillation conditions on initial vorticity.

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Works this paper leans on

57 extracted references · 57 canonical work pages · cited by 1 Pith paper · 2 internal anchors

  1. [1]

    Albritton, E

    D. Albritton, E. Bru´ e, and M. Colombo. Non-uniqueness of Leray solutions of the forced Navier-Stokes equations. Ann. of Math. (2) , 196(1):415–455, 2022

    work page 2022

  2. [2]

    Albritton, E

    D. Albritton, E. Bru´ e, M. Colombo, C. De Lellis, V. Giri, M. Janisch, and H. Kwon. Instability and non- uniqueness for the 2D Euler equations, after M. Vishik . Annals of Mathematics Studies. Princeton University Press, Princeton, 2024. In-text citations refer to the arXiv version: arXiv:2112.04943

    work page arXiv 2024

  3. [3]

    Bressan and R

    A. Bressan and R. Murray. On self-similar solutions to the incompressible Euler equations. J. Differential Equa- tions, 269(6):5142–5203, 2020

    work page 2020

  4. [4]

    Bressan and W

    A. Bressan and W. Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete Contin. Dyn. Syst. , 41(1):113–130, 2021

    work page 2021

  5. [5]

    Bru´ e and M

    E. Bru´ e and M. Colombo. Nonuniqueness of solutions to the Euler equations with vorticity in a Lorentz space. Comm. Math. Phys. , 403(2):1171–1192, 2023

    work page 2023

  6. [6]

    Buck and S

    M. Buck and S. Modena. Non-uniqueness and energy dissipation for 2D Euler equations with vorticity in Hardy spaces. arXiv:2306.05948, 2023

    work page arXiv 2023

  7. [7]

    Buckmaster, C

    T. Buckmaster, C. De Lellis, L. Sz´ ekelyhidi, Jr., and V. Vicol. Onsager’s conjecture for admissible weak solutions. Comm. Pure Appl. Math. , 72(2):229–274, 2019

    work page 2019

  8. [8]

    Buckmaster, N

    T. Buckmaster, N. Masmoudi, M. Novack, and V. Vicol. Intermittent convex integration for the 3D Euler equa- tions, volume 217 of Annals of Mathematics Studies . Princeton University Press, Princeton, NJ, [2023] ©2023

    work page 2023

  9. [9]

    Buckmaster and V

    T. Buckmaster and V. Vicol. Nonuniqueness of weak solutions to the Navier-Stokes equation. Ann. of Math. (2) , 189(1):101–144, 2019

    work page 2019

  10. [10]

    Bulut, M

    A. Bulut, M. K. Huynh, and S. Palasek. Convex integration above the Onsager exponent for the forced Euler equations. arXiv:2301.00804, 2023. A PROOF OF VISHIK’S NONUNIQUENESS THEOREM FOR THE FORCED 2D EULER EQUATION 31

    work page arXiv 2023

  11. [11]

    Castro, D

    A. Castro, D. C´ ordoba, and D. Faraco. Mixing solutions for the Muskat problem.Invent. Math., 226(1):251–348, 2021

    work page 2021

  12. [12]

    Castro and D

    A. Castro and D. Lear. Traveling waves near Couette flow for the 2D Euler equation. Comm. Math. Phys. , 400(3):2005–2079, 2023

    work page 2005

  13. [13]

    Cheskidov, P

    A. Cheskidov, P. Constantin, S. Friedlander, and R. Shvydkoy. Energy conservation and onsager’s conjecture for the euler equations. Nonlinearity, 21(6):1233, 2008

    work page 2008

  14. [14]

    Cheskidov and X

    A. Cheskidov and X. Luo. Sharp nonuniqueness for the Navier-Stokes equations. Invent. Math., 229(3):987–1054, 2022

    work page 2022

  15. [15]

    Constantin, W

    P. Constantin, W. E, and E. S. Titi. Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Comm. Math. Phys. , 165(1):207–209, 1994

    work page 1994

  16. [16]

    Cordoba, D

    D. Cordoba, D. Faraco, and F. Gancedo. Lack of uniqueness for weak solutions of the incompressible porous media equation. Arch. Ration. Mech. Anal., 200(3):725–746, 2011

    work page 2011

  17. [17]

    C´ ordoba and L

    D. C´ ordoba and L. Mart´ ınez-Zoroa. Blow-up for the incompressible 3D-Euler equations with uniformC1, 1 2 −ϵ ∩L2 force. arXiv:2309.08495, 2023

    work page arXiv 2023

  18. [18]

    C´ ordoba, L

    D. C´ ordoba, L. Mart´ ınez-Zoroa, and F. Zheng. Finite time singularities to the 3D incompressible Euler equations for solutions in C ∞(R3 \ {0}) ∩ C1,α ∩ L2. arXiv:2308.12197, 2023

    work page arXiv 2023

  19. [19]

    Dai and S

    M. Dai and S. Friedlander. Non-uniqueness of forced active scalar equations with even drift operators. arXiv:2311.06064, 2023

    work page arXiv 2023

  20. [20]

    De Lellis and L

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

    work page 2009

  21. [21]

    R. J. DiPerna and A. J. Majda. Concentrations in regularizations for 2-D incompressible flow. Comm. Pure Appl. Math., 40(3):301–345, 1987

    work page 1987

  22. [22]

    P. G. Drazin and W. H. Reid. Hydrodynamic stability. Cambridge University Press, Cambridge, 2004

    work page 2004

  23. [23]

    T. Elgindi. Finite-time singularity formation for C1,α solutions to the incompressible Euler equations on R3. Ann. of Math. (2) , 194(3):647–727, 2021

    work page 2021

  24. [24]

    V. Elling. Algebraic spiral solutions of the 2d incompressible Euler equations. Bull. Braz. Math. Soc. (N.S.) , 47(1):323–334, 2016

    work page 2016

  25. [25]

    Enciso, J

    A. Enciso, J. Pe˜ nafiel-Tom´ as, and D. Peralta-Salas. An extension theorem for weak solutions of the 3d incom- pressible Euler equations and applications to singular flows. arXiv:2404.08115, 2024

    work page arXiv 2024

  26. [26]

    Engel and R

    K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations , volume 194 of Graduate Texts in Mathematics . Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt

    work page 2000

  27. [27]

    G. L. Eyink. Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer. Phys. D, 78(3-4):222–240, 1994

    work page 1994

  28. [28]

    F¨ orster and L

    C. F¨ orster and L. Sz´ ekelyhidi, Jr. Piecewise constant subsolutions for the Muskat problem.Comm. Math. Phys. , 363(3):1051–1080, 2018

    work page 2018

  29. [29]

    Gancedo, A

    F. Gancedo, A. Hidalgo-Torn´ e, and F. Mengual. Dissipative Euler flows originating from circular vortex filaments. arXiv:2404.04250, 2024

    work page arXiv 2024

  30. [30]

    Garc´ ıa and J

    C. Garc´ ıa and J. G´ omez-Serrano. Self-similar spirals for the generalized surface quasi-geostrophic equations. arXiv:2207.12363, 2022. To appear in J. Eur. Math. Soc

    work page arXiv 2022

  31. [31]

    Gebhard, J

    B. Gebhard, J. J. Kolumb´ an, and L. Sz´ ekelyhidi. A new approach to the Rayleigh-Taylor instability. Arch. Ration. Mech. Anal., 241(3):1243–1280, 2021

    work page 2021

  32. [32]

    V. Giri, H. Kwon, and M. Novack. The L3-based strong Onsager theorem. arXiv:2305.18509, 2023

    work page arXiv 2023

  33. [33]

    Giri and R.-O

    V. Giri and R.-O. Radu. The 2D Onsager conjecture: a Newton-Nash iteration. arXiv:2305.18105, 2023

    work page arXiv 2023

  34. [34]

    Guillod and V

    J. Guillod and V. ˇSver´ ak. Numerical investigations of non-uniqueness for the Navier-Stokes initial value problem in borderline spaces. J. Math. Fluid Mech. , 25(3):Paper No. 46, 25, 2023

    work page 2023

  35. [35]

    N. Gunther. On the motion of fluid in a moving container. Izvestia Akad. Nauk USSR, Ser. Fiz.-Mat. , 20:1323– 1348, 1927

    work page 1927

  36. [36]

    H¨ older.¨Uber die unbeschr¨ ankte Fortsetzbarkeit einer stetigen ebenen Bewegung in einer unbegrenzten inkom- pressiblen Fl¨ ussigkeit.Math

    E. H¨ older.¨Uber die unbeschr¨ ankte Fortsetzbarkeit einer stetigen ebenen Bewegung in einer unbegrenzten inkom- pressiblen Fl¨ ussigkeit.Math. Z., 37(1):727–738, 1933

    work page 1933

  37. [37]

    P. Isett. A proof of Onsager’s conjecture. Ann. of Math. (2) , 188(3):871–963, 2018

    work page 2018

  38. [38]

    Jia and V

    H. Jia and V. ˇSver´ ak. Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions. Invent. Math., 196(1):233–265, 2014

    work page 2014

  39. [39]

    Jia and V

    H. Jia and V. ˇSver´ ak. Are the incompressible 3d Navier-Stokes equations locally ill-posed in the natural energy space? J. Funct. Anal., 268(12):3734–3766, 2015

    work page 2015

  40. [40]

    T. Kato. Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition

    work page 1995

  41. [41]

    Kiselev and V

    A. Kiselev and V. ˇSver´ ak. Small scale creation for solutions of the incompressible two-dimensional Euler equation. Ann. of Math. (2) , 180(3):1205–1220, 2014. 32 ´ANGEL CASTRO, DANIEL FARACO, FRANCISCO MENGUAL, AND MARCOS SOLERA

    work page 2014

  42. [42]

    Koch and D

    H. Koch and D. Tataru. Well-posedness for the Navier-Stokes equations. Adv. Math., 157(1):22–35, 2001

    work page 2001

  43. [43]

    Lichtenstein

    L. Lichtenstein. ¨Uber einige Existenzprobleme der Hydrodynamik homogener, unzusammendr¨ uckbarer, rei- bungsloser Fl¨ ussigkeiten und die Helmholtzschen Wirbels¨ atze.Math. Z., 23(1):89–154, 1925

    work page 1925

  44. [44]

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

    work page 2002

  45. [45]

    F. Mengual. Non-uniqueness of admissible solutions for the 2D Euler equation with Lp vortex data. arXiv:2304.09578, 2023

    work page arXiv 2023

  46. [46]

    Mengual and L

    F. Mengual and L. Sz´ ekelyhidi, Jr. Dissipative Euler flows for vortex sheet initial data without distinguished sign. Comm. Pure Appl. Math. , 76(1):163–221, 2023

    work page 2023

  47. [47]

    Novack and V

    M. Novack and V. Vicol. An intermittent Onsager theorem. Invent. Math., 233(1):223–323, 2023

    work page 2023

  48. [48]

    L. D. Rosa and J. Park. No anomalous dissipation in two-dimensional incompressible fluids. arXiv:2403.04668, 2024

    work page arXiv 2024

  49. [49]

    Scheffer

    V. Scheffer. An inviscid flow with compact support in space-time. J. Geom. Anal., 3(4):343–401, 1993

    work page 1993

  50. [50]

    F. Shao, D. Wei, and Z. Zhang. Self-similar algebraic spiral solution of 2-D incompressible Euler equations. arXiv:2305.05182, 2023

    work page arXiv 2023

  51. [51]

    Shnirelman

    A. Shnirelman. On the nonuniqueness of weak solution of the Euler equation. Comm. Pure Appl. Math. , 50(12):1261–1286, 1997

    work page 1997

  52. [52]

    Sz´ ekelyhidi

    L. Sz´ ekelyhidi. Weak solutions to the incompressible Euler equations with vortex sheet initial data.C. R. Math. Acad. Sci. Paris, 349(19-20):1063–1066, 2011

    work page 2011

  53. [53]

    Sz´ ekelyhidi, Jr

    L. Sz´ ekelyhidi, Jr. Relaxation of the incompressible porous media equation. Ann. Sci. ´Ec. Norm. Sup´ er. (4), 45(3):491–509, 2012

    work page 2012

  54. [54]

    M. Vishik. Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompress- ible fluid. Part I. arXiv:1805.09426, 2018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  55. [55]

    M. Vishik. Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompress- ible fluid. Part II. arXiv:1805.09440, 2018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  56. [56]

    Wolibner

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

    work page 1933

  57. [57]

    Yudovich

    V. Yudovich. Non-stationary flow of an ideal incompressible liquid. USSR Computational Mathematics and Mathematical Physics, 3(6):1407–1456, 1963. ´Angel Castro Instituto de Ciencias Matem´aticas, CSIC-UAM-UC3M-UCM 28049 Madrid, Spain E-mail address: angel castro@icmat.es Daniel Faraco Departamento de Matem´aticas, Universidad Aut´onoma de Madrid, Institu...

    work page 1963