Global well-posedness for the incompressible Euler equations in an endpoint Sobolev space

In-Jee Jeong; Rapha\"el Danchin

REVIEW 2 major objections 3 minor 45 references

Global well-posedness for the incompressible Euler equations in an endpoint Sobolev space

T0 review · 2 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read The paper proves that the vorticity of two-dimensional Euler flows, and of three-dimensional axisymmetric swirl-free Euler flows, keeps its endpoint critical Sobolev regularity W^{d,1} for all time, in contrast to the strong ill-posedness k

desk verdict Genuine new endpoint estimates for 2D and 3D axisymmetric Euler vorticity, but the 3D existence theorem is not proven as written. read the letter →

arxiv 2607.17110 v1 pith:6R4F7SP2 submitted 2026-07-19 math.AP

Raphaël Danchin , In-Jee Jeong This is my paper · ORCID

classification math.AP MSC 76B4735Q3535B40

keywords incompressibleEulerequationsvorticitywell-posednessSobolevspacesaxisymmetricflowsLorentzBesovendpointcritical

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

The paper establishes that the endpoint critical Sobolev space W^{d,1} of the vorticity is globally well-posed for the incompressible Euler equations: in two dimensions, any initial vorticity in W^{2,1} keeps that regularity for all time; in three dimensions, any axisymmetric vorticity without swirl in W^{3,1} also keeps it for all time. This settles the p=1 endpoint of the Sobolev scale W^{d/p,p} for these flow classes, the only critical Sobolev space where the velocity is Lipschitz, and it contrasts sharply with the strong ill-posedness results valid for every 1

What carries the argument

The machinery rests on three pieces. The first is the transport structure: in two dimensions the vorticity itself is advected, while in three-dimensional axisymmetric no-swirl flows the quantity α=ω_θ/r is advected, making its L^{p,1} norms time-independent. The second is the endpoint embedding: W^{d,1} embeds continuously into the Besov space B^0_{∞,1} (through Lorentz L^{d,1} spaces), and this Besov space controls the Lipschitz norm of the Biot–Savart velocity, giving the estimates needed to close Grönwall arguments. The third is a set of derivative equivalences between Cartesian and cylindrical coordinates, which convert W^{3,1} of an axisymmetric vorticity field into sums of cylindrical

What would settle it

A concrete way to test the claim: take a sequence of smooth axisymmetric no-swirl initial data with uniformly bounded W^{3,1} norms and check whether the third derivatives of the vorticity converge strongly in L¹ as the mollifier is removed. If a limit is only a bounded measure rather than an L¹ function—or if, for some data, the a priori bound on ∇³ω in L¹ blows up in finite time—the global propagation claim would be false.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the endpoint regularity W^{d,1} of the vorticity is propagated for all times: Theorem 1.1 for d=2 and Theorem 1.2 for three-dimensional axisymmetric flows without swirl. In the axisymmetric case, the vorticity is ω=ω_θ e_θ, and the ratio α=ω_θ/r solves a pure transport equation, ∂_t α+u·∇α=0, so its L^{3,1} norm is conserved; this removes the vortex-stretching obstruction. The authors then bootstrap: from the conserved α and the embedding W^{3,1}→B^0_{∞,1}, they control the Lipschitz velocity norm, then ∇ω in L^{3,1}, then ∇²ω in L^{3/2,1}, and finally ∇³ω in L¹, with double-exponential growth in time. The same bootstrap in two dimensio

Load-bearing premise

The existence part of the 3D theorem rests on an omitted compactness argument: the paper asserts, in Section 3, that the a priori estimates for smooth axisymmetric data pass to the limit and produce a solution with ∇³ω∈L¹, despite L¹ not being weakly compact; if this passage fails, Theorem 1.2's existence claim collapses.

Editorial extensions

If this is right

In two dimensions, every initial vorticity in W^{2,1}(R²) generates a unique global solution with ω∈C(R₊;W^{2,1}), with at most double-exponential growth of the W^{2,1} norm.
In three dimensions, every axisymmetric no-swirl initial vorticity in W^{3,1}(R³) generates a unique global solution with ∇³ω∈C(R₊;L¹); the L¹ norm of ∇³ω and the L^{3/2,1} norm of ∇²ω grow at most double exponentially.
The ill-posedness mechanisms that operate in W^{d/p,p} for 1<p<∞ cannot be transplanted to p=1, because at p=1 the velocity is Lipschitz; the endpoint case is therefore well-posed rather than ill-posed for these flow classes.
The axisymmetric no-swirl class is the one used in several ill-posedness constructions; this result shows that those constructions stop working exactly at the p=1 endpoint.
Global regularity in W^{3,1} for general, non-axisymmetric three-dimensional vorticity remains open; the authors identify the propagation of ∇³ω∈L¹ as the core difficulty.

Reading between the lines

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

If the p=1 endpoint is globally well-posed for these classes while every 1<p<∞ is strongly ill-posed, then the Sobolev-scale picture for Euler is non-monotone in p: the endpoint is the well-posed side of the cliff, not part of the ill-posed regime.
The double-exponential bounds are likely an artifact of the Grönwall iteration; a natural test is to examine concrete axisymmetric data, e.g. compactly supported near the axis, to see whether the third-derivative L¹ norm actually grows only exponentially.
A natural next step, suggested but not taken by the authors, is to extend the argument to non-axisymmetric 3D data by treating ∇³ω in the Besov space B^0_{1,∞} instead of L¹; endpoint product estimates make this plausible.
One could test the sharpness of the Lorentz-space framework by checking whether a slightly larger endpoint space, such as W^{3,p} with p close to 1, still admits global propagation or already exhibits the ill-posedness seen for p>1.

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.

Desk Editor's Note

read the letter

The real content here is a clean set of a priori estimates in the endpoint Sobolev spaces. For the 2D vorticity equation, the paper proves global W^{2,1} propagation, building on Cozzi–Harrison's local existence and Vishik's Besov control. That argument is straightforward and, as far as I can see, correct: the local solution plus the global bounds force T* = ∞. This alone is a nice complement to Bourgain–Li, who excluded p = 1.

The 3D axisymmetric no-swirl case is more interesting. The structural reduction to α = ω/r being transported, the use of Lorentz spaces L^{3,1} and L^{3/2,1}, and the careful tracking of the stretching terms in Section 3 are genuinely new. The double-exponential growth bounds for ∇³ω in L¹ are plausible and the componentwise estimates hang together. If the existence part can be completed, this settles the endpoint case for axisymmetric data in a way that the Bourgain–Li construction cannot touch.

But the existence part is not completed. The paper explicitly says the passage from smooth data to an L¹-critical solution "can be overcome by following faithfully the approach of [15] for the 2D case, and thus omitted." That is a load-bearing sentence. L¹ is not weakly compact, so a uniform bound on ∇³ω_n in L¹ does not give you an L¹ function in the limit; you only get a measure. The 2D compactness argument in Cozzi–Harrison is written for the scalar transport equation, while the 3D axisymmetric system has the stretching term vωθ and third-order terms involving α, ∇α, and ∇²α. It is not automatic that the 2D proof transfers. Since uniqueness in the Besov class is already available from Abidi–Hmidi–Keraani and Danchin, there is a path: show the mollified solutions converge in a suitable weak sense to the unique Besov solution, then prove via some continuity argument that the uniform W^{3,1} bounds force the limit to have the claimed regularity. But that argument is not here. The a priori estimates alone do not prove Theorem 1.2 as stated.

This is not a fatal flaw in the estimates, and the authors are honest about the gap. The paper is worth a serious referee: the 2D theorem appears complete, and the 3D estimates are valuable enough that a referee report asking for the missing compactness argument would be a reasonable outcome. I would cite the a priori estimates in my own work, and I would bring the paper to a reading group. But I would not yet cite Theorem 1.2 as a proven global well-posedness result.

Referee Report

2 major / 3 minor

Summary. The paper studies the incompressible Euler equations in the endpoint critical Sobolev spaces W^{2,1}(R^2) and W^{3,1}(R^3) for the vorticity. In two dimensions it proves global propagation of W^{2,1} vorticity regularity, combining the Cozzi–Harrison local existence theorem with Vishik's B^0_{\infty,1} control and Gronwall-type estimates; the vorticity norm is shown to grow at most double exponentially. In three dimensions, for axisymmetric flows without swirl, it claims global propagation of W^{3,1} vorticity regularity, with global existence and uniqueness of a solution in C(R_+; W^{3,1}) and additional Lorentz/Besov regularity. The 3D proof is organized as a sequence of a priori estimates: first \nabla\omega \in L^{3,1}, then \nabla^2\omega \in L^{3/2,1}, and finally \nabla^3\omega \in L^1, using the transport structure of \alpha = \omega_\theta/r and a detailed set of cylindrical-to-Cartesian norm equivalences. The main technical gap is that the existence part of the 3D theorem is not actually proved: the passage from mollified smooth data to a solution with L^1 third derivatives is explicitly deferred in Section 3.

Significance. If the existence gap is filled, the result would be significant: it would settle the endpoint case p=1 of the critical Sobolev spaces W^{d/p,p}, in stark contrast to the Bourgain–Li strong ill-posedness for all 1<p<\infty. The paper supplies explicit exponential/double-exponential a priori bounds, uses sharp Lorentz-space embeddings, and carefully relates Cartesian and cylindrical derivative norms. The 2D part is essentially sound and rests on a clean combination of known theorems. The 3D a priori estimates are plausible and the structure exploited (transport of \alpha) is natural. However, because a load-bearing compactness passage is omitted, Theorem 1.2 as stated is not proven in the manuscript.

major comments (2)

[Section 3, existence part of Theorem 1.2] The existence part of Theorem 1.2 is not proved. After smoothing the data, the text says the L^1 weak-* compactness difficulty 'can be overcome by following faithfully the approach of [15] for the 2D case, and thus omitted.' This is load-bearing: a uniform W^{3,1} bound on mollified solutions only gives a measure-valued limit for the third derivatives, since L^1(R^3) is not weakly compact and is not a dual space. More is needed to ensure that the limit is a function with ∇^3\omega \in C(R_+;L^1). The 2D argument in [15] is specific to the scalar transport equation (1.2), whereas the axisymmetric system (1.5) contains the stretching term v\omega_\theta and lower-order terms in \alpha. Moreover, no local well-posedness theorem in W^{3,1} is cited, so for arbitrary W^{3,1} data the a priori estimates in §3.1–3.3 do not yet apply to an existing solution. The theorem as stated ('admits a uniq
[Section 3.3, equation for D_t∂^3_{zrr}ωθ] In the displayed equation for D_t∂^3_{zrr}\omega_\theta, a commutator term is missing. Writing f=∂^2_{rr}\omega_\theta, from (3.8) one has D_t∂_z f = ∂_z(D_t f) - ∂_z u·∇f. The term -∂_z u·∇∂^2_{rr}\omega_\theta does not appear in the displayed formula, nor is it accounted for in the subsequent estimates. This is a genuine algebraic gap in the derivation of the central third-derivative bound. The missing term is of the same type as the other third-order transport terms and can be bounded by \|∂_z u\|_\infty \|∇∂^2_{rr}\omega_\theta\|_{L^1}, which is controlled by the already available single-exponential factor; thus the final inequality is likely unaffected. Nevertheless, the displayed equation should be corrected.

minor comments (3)

[Section 3.3, paragraph 'Bounding ∂^3_{zzr}ωθ'] The text says 'Differentiating (3.6) once with respect to z' but the displayed equation is for ∂^3_{zzr}\omega_\theta, which requires two derivatives with respect to z. Please clarify the wording.
[Throughout] There are a few typographical issues: 'illposedness' is sometimes written without a hyphen, and reference [31] contains a typo ('Helmoltz' instead of 'Helmholtz'). These do not affect the mathematics.
[Section 3.2, displayed estimate for ∇^2ωθ] In the paragraph following (3.4), the term \|∂^2_{zz}(r^{-1}u_r)\omega_\theta\|_{L^{3/2,1}} is bounded by a product involving \|∇^3u\|_{L^{3,1}}\|\omega_\theta\|_\infty; the intended Lorentz-space Hölder inequality should be stated so that the reader can verify the exponent (the product lands in L^{3/2,1/2} ⊂ L^{3/2,1}).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the global bounds are derived from external theorems and new Gronwall estimates; the only flagged item is an omitted L^1 compactness passage in 3D, which is a gap, not a circular reduction.

full rationale

The claimed derivation chain is not circular. In 2D, Theorem 1.1 rests on the external local well-posedness theorem of Cozzi–Harrison [15], Vishik/Hmidi–Keraani propagation in B^0_{\infty,1}, and elementary transport/Biot–Savart estimates (Section 2); the Gronwall arguments (2.5) are new bounds, not restatements of the assumptions. In 3D, the existence and uniqueness of the underlying axisymmetric solution is imported from [1,16]; [16] is a self-citation, but it is co-cited with the independent [1] and is used as a published theorem (including the bound (3.1)), so it does not smuggle in the target W^{3,1} conclusion. The chain in Sections 3.1–3.3 then consists of direct estimates on the transport equations for \nabla\omega, \nabla^2\omega, and each third derivative of \omega_\theta, closed by Gronwall and the commutator identity (3.7). I find no step in which the conclusion is defined in terms of the hypothesis, no fitted quantity is renamed as a prediction, and no external result's assumptions include the target theorem. The passage that deserves flagging is the deferred compactness step in Section 3: 'since the space L^1 is not stable by weak * compactness, it may happen that the third order derivatives of the vorticity are only bounded measures rather than nice L^1 functions. This can be overcome by following faithfully the approach of [15] for the 2D case, and thus omitted.' This is a genuine proof gap for the existence part of Theorem 1.2 (L^1(R^3) is not weakly compact, so the a priori bounds do not alone yield \nabla^3\omega\in C(R_+;L^1)), but it is an omitted argument, not an equivalence-by-construction; it does not make the derivation circular. Hence the score is 1 rather than higher.

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

The paper's central claims rest on a package of prior theorems and standard harmonic-analysis facts rather than on freely fitted constants or invented entities. The most important external inputs are Cozzi-Harrison's local well-posedness in W^{2,1}(R²), Vishik's and Hmidi-Keraani's global results in critical Besov spaces, and Danchin's/Abidi-Hmidi-Keraani's global theory for axisymmetric flows with α ∈ L^{3,1}. The standard facts include Sobolev-Lorentz embeddings, real interpolation, Riesz-transform boundedness, and norm equivalences for axisymmetric fields. No free parameters or invented entities are used.

assumptions (5)

domain assumption Local well-posedness of 2D Euler in W^{2,1}(R²) (Cozzi-Harrison [15])
Used in §2 to obtain a unique maximal solution and the blow-up criterion limsup ∥ω(t)∥_{W^{2,1}} = ∞ at T*. The global result of this paper builds on it.
domain assumption Global well-posedness for 2D Euler in critical Besov spaces B^{2/p}_{p,1}, including p=∞ (Vishik [41], Hmidi-Keraani [23])
Used in §2 to control ∥ω(t)∥_{B^0_{∞,1}} globally, which in turn controls ∥∇u∥_{L∞}.
domain assumption Global well-posedness for axisymmetric Euler without swirl with α ∈ L^{3,1} (Abidi-Hmidi-Keraani [1], Danchin [16])
Used in §3 to have a global unique solution in C(R+; B^0_{∞,1}) once ω0 ∈ B^0_{∞,1} and α0 ∈ L^{3,1}.
standard math Embeddings and operator bounds for Lorentz/Besov spaces (isoperimetric inequality, Riesz transform boundedness, real interpolation; Appendix A)
Used throughout: W^{1,1} → L^{d',1}, L^{p,1} → B^{d/q-d/p}_{q,1}, and ∇²Δ^{-1} maps L^{p,1} to itself.
standard math Norm equivalences for axisymmetric fields (Appendix B, Lemma B.1/B.2)
Used to rewrite Cartesian derivatives of u and ω in terms of (r,z) derivatives and r^{-1} terms; derived from pointwise identities.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Global well-posedness for the incompressible Euler equations in an endpoint Sobolev space." pith.science (2026). https://pith.science/paper/6R4F7SP2

@misc{pith2026260717110,
  author       = {Pith},
  title        = {Pith review of: Global well-posedness for the incompressible Euler equations in an endpoint Sobolev space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6R4F7SP2}},
  note         = {Machine review of arXiv:2607.17110}
}

abstract

We consider the initial value problem for the vorticity equation in the endpoint critical Sobolev space $W^{d,1}(\mathbb{R}^{d})$ for $d = 2, 3$. In two dimensions, we prove global propagation of the $W^{2,1}(\mathbb{R}^{2})$ regularity of the vorticity. In three dimensions, for axisymmetric flows without swirl, we propagate $W^{3,1}(\mathbb{R}^{3})$ regularity of the vorticity for all times. These are in stark contrast to existing strong ill-posedness results in critical Sobolev spaces $W^{d/p,p}(\mathbb{R}^{d})$ for all $1 < p < \infty$, which were based on axisymmetric flows without swirl when $d = 3$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

45 extracted references · 5 linked inside Pith

[15]

Cozzi and N

E. Cozzi and N. Harrison: Local existence for the 2D Euler equations in a critical Sobolev space, Nonlinear Anal.,261(2025), Paper No. 113846, 14 pp

2025
[1]

Abidi, T

H. Abidi, T. Hmidi and S. Keraani: On the global well-posedness for the axisymmetric Euler equations, Math. Ann.,347(2010), no. 1, 15–41

2010
[2]

Bang and A

J. Bang and A. Cheskidov: Sharp Ill-Posedness of the Euler Equations in Lorentz Spaces, arXiv:2605.16502

arXiv
[3]

Bahouri, J.-Y

H. Bahouri, J.-Y. Chemin and R. Danchin:Fourier Analysis and Nonlinear Partial Differential Equa- tions,Grundlehren der mathematischen Wissenschaften,343, Springer (2011)

2011
[4]

Bardos, E

C. Bardos, E. S. Titi: Loss of smoothness and energy conserving rough weak solutions for the 3d Euler equations,Discrete Contin. Dyn. Syst. Ser. S,3(2):185–197, 2010

2010
[5]

Beale, J

J.T. Beale, J. T. Kato and A. Majda: Remarks on the breakdown of smooth solutions for the 3-D Euler equations,Comm. Math. Phys.,94(1984), no. 1, 61–66

1984
[6]

Bergh and J

J. Bergh and J. L¨ ofstr¨ om:Interpolation spaces. An introduction., Grundlagen der mathematischen Wissenschaften,223, Springer, Berlin-New York, 1976

1976
[7]

Bourgain and D

J. Bourgain and D. Li: Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces,Invent. Math.,201(2015) 97–157

2015

Show all 45 references

[8]

Bourgain and D

J. Bourgain and D. Li: Strong ill-posedness of the incompressible Euler equations in integerC m spaces, Geom. Funct. Anal.,25(1) (2015) 1–86

2015
[9]

Bourgain and D

J. Bourgain and D. Li: Strong ill-posedness of the 3D incompressible Euler equation in borderline spaces,Int. Math. Res. Not.,2021, no. 16, 12155–12264

2021
[10]

Chae: Local existence and blow-up criterion for the Euler equations in the Besov spaces,Asymptot

D. Chae: Local existence and blow-up criterion for the Euler equations in the Besov spaces,Asymptot. Anal.38(2004), no. 3-4, 339–358

2004
[11]

Chemin: Perfect Incompressible Fluids, vol

J.-Y. Chemin: Perfect Incompressible Fluids, vol. 14, Oxford University Press, 1998

1998
[12]

Cheskidov and R

A. Cheskidov and R. Shvydkoy: Ill-posedness of the basic equations of fluid dynamics in Besov spaces, Proc. Amer. Math. Soc.,138(2010), no. 3, 1059–1067

2010
[13]

Q. Chen, C. Miao and Z. Zhang: On the well-posedness of the ideal MHD equations in the Triebel–Lizorkin spaces,Arch. Ration. Mech. Anal.,195(2010), 561–578

2010
[14]

C´ ordoba, L

D. C´ ordoba, L. Mart ´ ınez-Zoroa and W. O˙ za´ nski: Instantaneous gap loss of Sobolev regularity for the 2D incompressible Euler equations,Duke Math. J.173(10) 1931–1971

1931
[16]

Danchin: Axisymmetric incompressible flows with bounded vorticity,Uspekhi Mat

R. Danchin: Axisymmetric incompressible flows with bounded vorticity,Uspekhi Mat. Nauk,62(2007), no. 3(375), 73–94; translation inRussian Math. Surveys,62(2007), no. 3, 475–496. 14

2007
[17]

Elgindi: Finite-time singularity formation forC 1,α solutions to the incompressible Euler equations onR 3,Ann

T. Elgindi: Finite-time singularity formation forC 1,α solutions to the incompressible Euler equations onR 3,Ann. of Math.,194(2021), no. 3, 647–727

2021
[18]

Elgindi and I.-J

T. Elgindi and I.-J. Jeong: Ill-posedness for the incompressible Euler equations in critical Sobolev spaces,Ann. PDE,3(2017), no. 1, Paper No. 7, 19 pp

2017
[19]

Elgindi and I.-J

T. Elgindi and I.-J. Jeong: On singular vortex patches, I: Well-posedness issues,Memoirs of the Amer. Math. Soc.,283(2023), no. 1400, v+89 pp

2023
[20]

T. M. Elgindi and N. Masmoudi:L ∞ ill-posedness for a class of equations arising in hydrodynamics, Arch. Ration. Mech. Anal.,235(2020), no. 3, 1979–2025

2020
[21]

T. M. Elgindi, R.W. Murray and A. Said: Wellposedness and singularity formation beyond the Yu- dovich class, to appear inJ. of the European Math. Society
[22]

Gunther: On the motion of fluid in a moving container,Izvestia Akad

N. Gunther: On the motion of fluid in a moving container,Izvestia Akad. Nauk USSR, Ser. Fiz.-Mat., Vol. 20, 1927, pp.1323-1348, 1503-1532; Vol. 21, 1927, pp. 521-526, 735-756; Vol. 22, 1928, pp. 9-30

1927
[23]

Hmidi and S

T. Hmidi and S. Keraani: Existence globale pour le syst` eme d’Euler incompressible 2-D dansB 1 ∞,1 , C. R. Math. Acad. Sci. Paris,341(2005), no. 11, 655–658

2005
[24]

Hwang and H.-C

J. Hwang and H.-C. Pak: Persistence of the solution to the Euler equations in an end-point critical Triebel-Lizorkin space.Math. Methods Appl. Sci.,48(2025), no. 2, 2421–2433

2025
[25]

Jeong: Loss of regularity for the 2D Euler equations,J

I.-J. Jeong: Loss of regularity for the 2D Euler equations,J. Math. Fluid Mech.,23(95) (2021) 339–358

2021
[26]

Jeong and J

I.-J. Jeong and J. Kim: Strong ill-posedness for SQG in critical Sobolev spaces,Anal. PDE,17(2024), no. 1, 133–170

2024
[27]

Jeong, L

I.-J. Jeong, L. Mart ´ ınez-Zoroa and W. S. O˙ za´ nski: Instantaneous continuous loss of Sobolev regularity for the 3D incompressible Euler equation, arXiv:2508.06333

arXiv
[28]

Jeong and B

I.-J. Jeong and B. Pausader: Discrete Schr¨ odinger equation and ill-posedness for the Euler equation, Discrete Contin. Dyn. Syst.,37(2017), no. 1, 281–293

2017
[29]

Jeong and T

I.-J. Jeong and T. Yoneda: Vortex stretching and enhanced dissipation for the incompressible 3D Navier-Stokes equations,Math. Ann.,380(2021), no. 3-4, 2041–2072

2021
[30]

Jo and J

M.J. Jo and J. Kim: Velocity blow-up inC 1 ∩H 2 for the 2D Euler equations, arXiv:2211.13872

arXiv
[31]

L. Lichtenstein: ¨Uber einige Existenzprobleme der Hydrodynamik homogenerunzusammendr¨ uckbarer, reibungsloser Fl¨ ußigkeiten und die Helmoltzochen Wirbelsatze,Mathematische Zeitschrift,23, 1925, 89–154 ;26, 1927, 196–323 ;28, 1928, pages 387–415 and32, 1930, 608–725

1925
[32]

Luo: Illposedness of incompressible fluids in supercritical Sobolev spaces, arXiv: 2404.07813

X. Luo: Illposedness of incompressible fluids in supercritical Sobolev spaces, arXiv: 2404.07813

arXiv
[33]

Kato and G

T. Kato and G. Ponce: Well-posedness of the Euler and Navier-Stokes equations in the Lebesgue spaces Lp s(R2),Rev. Mat. Iberoamericana,2(1986), no. 1-2, 73–88

1986
[34]

Kato and G

T. Kato and G. Ponce: Commutator estimates and the Euler and Navier-Stokes equations,Comm. Pure Appl. Math.,41(1988), no. 7, 891–907

1988
[35]

Kim and I.-J

J. Kim and I.-J. Jeong: A simple ill-posedness proof for incompressible Euler equations in critical Sobolev spaces,J. Funct. Anal.,283(10) (2022)

2022
[36]

Misiolek and T

G. Misiolek and T. Yoneda: Continuity of the solution map of the Euler equations in H¨ older spaces and weak norm inflation in Besov spaces,Trans. Amer. Math. Soc.,370, (2018), 4709–4730

2018
[37]

Pak and Y.J

H.C. Pak and Y.J. Park: Existence of solution for the Euler equations in a critical Besov space B1 ∞,1(Rn),Comm. Partial Differential Equations,29(7-8) (2004) 1149–1166

2004
[38]

Pak and Y.J

H.C. Pak and Y.J. Park: Persistence of the incompressible Euler equations in a Besov spaceB d 1,1(Rd), Adv Differ Equ,2013, 153 (2013)

2013
[39]

Shikh Khalil: Strong ill-posedness of the 2d Incompressible Euler equation in critical Besov spaces, arXiv:2606.08924

K.R. Shikh Khalil: Strong ill-posedness of the 2d Incompressible Euler equation in critical Besov spaces, arXiv:2606.08924

arXiv
[40]

Ukhovski ` ı and V.I

M.R. Ukhovski ` ı and V.I. Yudovich: Axially symmetric flows of ideal and viscous fluids filling the whole space,J. Appl. Math. Mech.,32(1968) 52–61

1968
[41]

Vishik: Hydrodynamics in Besov spaces,Arch

M. Vishik: Hydrodynamics in Besov spaces,Arch. Ration. Mech. Anal.,145:3 (1998), 197–214

1998
[42]

Vishik: Incompressible flows on an ideal fluid with vorticity in borderline spaces of Besov type, Ann

M. Vishik: Incompressible flows on an ideal fluid with vorticity in borderline spaces of Besov type, Ann. Sci. Ecole Norm. Sup.,32(1999), No. 6, 769–812

1999
[43]

Wolibner: Un th´ eeor` eme sur l’existence du mouvement plan d’un fluide parfait, homog` ene, incom- pressible, pendant un temps infiniment long,Math

W. Wolibner: Un th´ eeor` eme sur l’existence du mouvement plan d’un fluide parfait, homog` ene, incom- pressible, pendant un temps infiniment long,Math. Z.,37(1933), no. 1, 698–726

1933
[44]

Yudovich: Non-stationary flow of an ideal incompressible liquid, U.S.S.R,Comput

V.I. Yudovich: Non-stationary flow of an ideal incompressible liquid, U.S.S.R,Comput. Math. Math. Phys.,3(6) (1963) 1407–1456

1963
[45]

Zhou: Local well-posedness for the incompressible Euler equations in the critical Besov spaces,Ann

Y. Zhou: Local well-posedness for the incompressible Euler equations in the critical Besov spaces,Ann. Inst. Fourier,54:3 (2004), 773–786. 15 ∗ Univ Paris Est Creteil, Univ Gustave Eiffel, CNRS, LAMA UMR8050, F-94010 Creteil, France. Email address:danchin@u-pec.fr School of Ma...

2004

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [15]

Cozzi and N

E. Cozzi and N. Harrison: Local existence for the 2D Euler equations in a critical Sobolev space, Nonlinear Anal.,261(2025), Paper No. 113846, 14 pp

2025

[2] [1]

Abidi, T

H. Abidi, T. Hmidi and S. Keraani: On the global well-posedness for the axisymmetric Euler equations, Math. Ann.,347(2010), no. 1, 15–41

2010

[3] [2]

Bang and A

J. Bang and A. Cheskidov: Sharp Ill-Posedness of the Euler Equations in Lorentz Spaces, arXiv:2605.16502

arXiv

[4] [3]

Bahouri, J.-Y

H. Bahouri, J.-Y. Chemin and R. Danchin:Fourier Analysis and Nonlinear Partial Differential Equa- tions,Grundlehren der mathematischen Wissenschaften,343, Springer (2011)

2011

[5] [4]

Bardos, E

C. Bardos, E. S. Titi: Loss of smoothness and energy conserving rough weak solutions for the 3d Euler equations,Discrete Contin. Dyn. Syst. Ser. S,3(2):185–197, 2010

2010

[6] [5]

Beale, J

J.T. Beale, J. T. Kato and A. Majda: Remarks on the breakdown of smooth solutions for the 3-D Euler equations,Comm. Math. Phys.,94(1984), no. 1, 61–66

1984

[7] [6]

Bergh and J

J. Bergh and J. L¨ ofstr¨ om:Interpolation spaces. An introduction., Grundlagen der mathematischen Wissenschaften,223, Springer, Berlin-New York, 1976

1976

[8] [7]

Bourgain and D

J. Bourgain and D. Li: Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces,Invent. Math.,201(2015) 97–157

2015