Sharp energy regularity and typicality results for H\"older solutions of incompressible Euler equations

Luigi De Rosa; Riccardo Tione

REVIEW 1 major objections 3 minor 10 references

Sharp energy regularity and typicality results for H\"older solutions of incompressible Euler equations

T0 review · 1 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read This paper establishes that, below the Hölder exponent $1/3$, weak solutions of the incompressible Euler equations with kinetic energy at exactly the sharp regularity $2\theta/(1-\theta)$ and no better Sobolev regularity are typical in a…

desk verdict Theorem 1.1 and 1.3 look solid, but Theorem 1.2 as written has a genuine parameter-ordering gap in the energy regularity; the paper deserves refereeing, not desk rejection. read the letter →

arxiv 1908.03529 v4 pith:PT32QSUL submitted 2019-08-08 math.AP

Luigi De Rosa , Riccardo Tione This is my paper · ORCID

classification math.AP MSC 35Q3135D3076B0326A21

keywords incompressibleEulerequationsHölderweaksolutionskineticenergyregularityconvexintegrationBairecategoryresidualset

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

Below the critical Hölder exponent $1/3$, weak solutions of the incompressible Euler equations can dissipate energy, and the kinetic energy $e_v$ of any $C^\theta$ solution is known to satisfy $e_v\in C^{\theta^*}$ with $\theta^* = 2\theta/(1-\theta)$. This paper proves that this bound is sharp in a strong generic sense: inside the space $X_\theta$ obtained by closing the smoother weak solutions in the $C^\theta$ norm, the set of solutions whose energy is in $C^{\theta^*}$ but in no better fractional Sobolev space $W^{\theta^*+\varepsilon,p}$ on any open time interval is residual, hence typical in the Baire category sense. The same machinery shows that smooth solutions form a nowhere dense set in the full space of all $C^\theta$ weak solutions. If the genericity statement could be transferred from $X_\theta$ to the full solution space, it would resolve the outstanding part of the sharp-energy-profile conjecture; the obstruction is a technical gap in starting the convex-integration iteration from merely $C^\theta$ data.

What carries the argument

The argument is carried by the convex-integration induction for the Euler–Reynolds system, the approximate system $\partial_t v_q + \operatorname{div}(v_q\otimes v_q)+\nabla p_q = \operatorname{div} \check R_q$. At each step one adds a highly oscillatory, divergence-free perturbation built from Mikado flows, which are exact periodic solutions of the Euler equations with prescribed Reynolds stress, while a time mollification of the target energy profile $e$ controls the energy gap in the perturbation step. The sharp exponent $\theta^* = 2\theta/(1-\theta)$ organizes the construction: the mollification scale is chosen from $e$'s $C^{\theta^*}$ norm, and the Baire-category argument writes the complement of $Y_\theta$ as a countable union of closed sets $C_{m,n,r,s}$, each shown to have empty interior by perturbing any candidate ball with a solution carrying an energy profile that violates the better-Sobolev condition.

What would settle it

Find one open ball in $X_\theta$ in which every solution has kinetic energy belonging to $W^{\theta^*+\varepsilon,p}$ on some fixed open interval for some $\varepsilon>0$ and $p\ge1$; Theorem 1.2 asserts no such ball exists, so any such ball would refute the residuality claim.

Watch

Extended reading notes

Core claim

The central discovery is that the sharp energy-regularity threshold is not a feature of carefully engineered counterexamples but a generic phenomenon. For every $\theta\in(0,1/3)$, the set $Y_\theta$ of solutions in $X_\theta$ whose kinetic energy $e_v$ belongs to $C^{\theta^*}([0,T])$ but fails to belong to $W^{\theta^*+\varepsilon,p}(I)$ for any $\varepsilon>0$, any $p\ge1$, and any open interval $I\subset[0,T]$ is residual in $X_\theta$. Equivalently, outside a meager set, the kinetic energy has exactly the maximal Hölder regularity permitted by the a priori bound and no extra fractional differentiability on any time interval. A companion result shows that every strictly positive profile $e\in C^{\theta^*+\gamma}([0,T])$ is realized by some $C^\theta$ weak solution, extending the earlier smooth-profile construction. Finally, the paper proves that smooth solutions are nowhere dense in the full space of $C^\theta$ weak solutions, so regularity is exceptional rather than typical.

Load-bearing premise

The load-bearing premise is that the ambient space is $X_\theta$, the $C^\theta$-closure of stronger weak solutions, and that the prescribed energy profile is strictly positive; if either fails, the typicality statement as proved may no longer hold.

Editorial extensions

If this is right

For every $\theta\in(0,1/3)$, there exist $C^\theta$ weak solutions whose kinetic energy belongs to $C^{\theta^*}$ but fails to lie in any $W^{\theta^*+\varepsilon,p}$ on any open interval; in particular the energy is not of bounded variation on any time interval.
Within $X_\theta$, such maximally irregular energy behavior is residual: almost every solution in the Baire sense has energy that cannot be improved above the sharp exponent.
Smooth solutions are nowhere dense in the space of all $C^\theta$ weak solutions, so at Hölder regularity below $1/3$ the smooth or even more regular solutions form a topologically negligible set.
Every strictly positive $C^{\theta^*+\gamma}$ energy profile is exactly realized by a $C^\theta$ weak solution, so the flexibility of convex integration extends to non-smooth, merely Hölder energy profiles.

Reading between the lines

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

If one could prove that every $C^\theta$ weak solution lies in the $C^\theta$-closure of smoother weak solutions, then $X_\theta$ would be the full solution space and the residuality statement would settle the original conjecture completely; Section 6 of the paper suggests the current obstruction is technical rather than a genuine counterexample.
The strict positivity hypothesis is a real boundary of the method: prescribing an energy profile that vanishes on an interval would require a new mechanism, since the inductive estimates force $e(t)\ge \delta_1\lambda_0^{-\alpha}>0$.
The same Baire framework could be tested for other critical exponents, such as Besov or $L^p$-based regularity scales at the energy-conservation threshold, since the appendix already converts spatial closeness into space-time regularity at the matching exponents.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Colleague: this paper has two results worth keeping and one that currently doesn't close. The solid pieces are Theorem 1.1, which adapts the Buckmaster–De Lellis–Székelyhidi–Vicol convex integration scheme to energy profiles that are only C^{θ*+γ} rather than smooth, and Theorem 1.3, which shows smooth solutions are nowhere dense in the full space of C^θ weak solutions. The proof of the inductive proposition is detailed, and the appendix reproving Isett's time-regularity lemma is a useful, self-contained addition. No circularity red flags; the self-citation to [2] is backed by a proof in the appendix.

The problem is Theorem 1.2. The stress-test note is right. In §3.2 the authors choose θ < θ′ < θ″ < β < η, which forces η* > (θ′)* because x ↦ 2x/(1−x) is increasing. The energy profile e(t) = ∫|u|² dx + (ρ/2)f(t) then combines an e_u in C^{(θ′)*} (by Isett's bound) with an f in C^{η*}. Since (θ′)* < η*, the sum is only C^{(θ′)*} in general, not C^{η*}. Yet Proposition 2.2 is applied with exactly that η, and the proof claims ‖e‖_{η*} ≤ E uniformly. For an arbitrary u in the closed set C_{m,n,r,s}, e_u need not be C^{η*}; Isett–Oh examples show it can sit exactly at the lower regularity. So the hypothesis of Proposition 2.2 is not met.

This is not a cosmetic slip. The genericity argument depends entirely on prescribing this e and getting the sharp failure of Sobolev regularity. The ordering θ′ < θ″ < β < η is forced if you want output regularity above θ′ while keeping β < η, so a repair would need a different mechanism, not just a parameter tweak.

Where does that leave the paper? Theorem 1.1 and Theorem 1.3 look sound and are genuinely useful extensions. The paper is transparent about the Xθ restriction and the strict positivity of the energy. But the advertised partial resolution of Isett–Oh Conjecture 1 rests on Theorem 1.2, and that proof currently has a load-bearing gap. I would send this to a serious referee, with the clear instruction to focus on whether Theorem 1.2 can be salvaged. If it can, this is a strong paper; as it stands, it is a solid Theorem 1.1 plus an unproved genericity claim.

Recommendation: accept for peer review, but expect major revision and be ready for the possibility that Theorem 1.2 is not repairable in its current form.

Referee Report

1 major / 3 minor

Summary. The paper studies Hölder-continuous weak solutions of the incompressible Euler equations on the three-dimensional torus for Hölder exponents θ<1/3. Theorem 1.1 constructs, for any strictly positive energy profile e∈C^{θ*+γ}([0,T]) with θ*=2θ/(1−θ), a C^θ weak solution whose kinetic energy equals e; this drops the smoothness assumption on e used in earlier convex-integration schemes. Theorem 1.2 states that in the space Xθ obtained as the C^θ closure of more regular weak solutions, the set of solutions whose energy lies exactly in C^{θ*} and in no higher fractional Sobolev space on any open interval is residual, partially resolving a conjecture of Isett and Oh. Theorem 1.3 states that smooth solutions are nowhere dense in the space of all C^θ weak solutions. The proofs adapt the Buckmaster–De Lellis–Székelyhidi–Vicol convex-integration iteration, mollifying the non-smooth energy profile at scale ε_q, and use a Baire-category argument. An appendix proves a spacetime regularity estimate for differences of weak solutions.

Significance. If the proof is completed, the paper would be a significant contribution: it establishes the sharp energy-regularity exponent 2θ/(1−θ) as a typical phenomenon for Hölder weak solutions below the Onsager threshold, and it extends the admissible energy profiles in the convex-integration construction from smooth profiles to Hölder profiles. Theorem 1.3 is a clean strengthening of the known meagerness of smooth solutions. The paper is also careful to explain the obstruction to working in the full space of C^θ solutions, namely the 'θ−β gap' discussed in Section 6. The appendix reproves the needed spacetime regularity lemma rather than merely citing it, which is a useful self-contained feature. However, the proof of Theorem 1.2 currently contains a load-bearing parameter-ordering gap that must be fixed before the central claim is established.

major comments (1)

[Section 3.2, Eq. (3.5) and application of Proposition 2.2] The parameter choices in the proof of Theorem 1.2 are internally inconsistent and, as stated, fail to verify the hypothesis e∈C^{η*} needed for Proposition 2.2. After assuming θ*<(θ')*<θ*+1/(2m), the authors fix θ<θ'<θ''<β<η with η*<θ*+1/(2m). Since the map x↦2x/(1−x) is increasing on (0,1/3), this gives (θ')*<η*. The energy profile in (3.5) is e(t)=∫_{T^3}|u|^2 dx+(ρ/2)f(t), where u∈C^{θ'}; by (1.2) one only knows e_u∈C^{(θ')*}, which is weaker than C^{η*}. The bound ‖ẽ‖_{η*}≲‖e_u‖_{η*}+‖f‖_{η*} used immediately before applying Proposition 2.2 is therefore unjustified: ‖e_u‖_{η*} may be infinite. Moreover, this ordering contradicts the later assertion 'since β<θ′' used in the verification of (3.13), since θ'<θ''<β would instead give β>θ'. The repair is to choose the parameters in the order θ<θ''<β<η<θ', with η*<θ*+1/(2m). Then (θ')*>η*, so e_u∈C^{(θ')*}⊂C^{η*}, and (3.13) is satisfiable because β<θ'. This ordering is compatible with Proposition 2.2 and with the requirement v∈C^{θ''}⊂Xθ. As written, however, the proof of the claim at the heart of Theorem 1.2 is not valid.

minor comments (3)

[Remark 3.1] Remark 3.1 opens with 'we fixed parameters 0<β<θ′<1/3', which conflicts with the ordering θ<θ'<θ''<β<η stated in the proof of Theorem 1.2. After the parameter ordering is corrected, this remark should be updated to be consistent with the proof.
[Section 3.3, proof of Theorem 1.3] In the proof of Theorem 1.3, the phrase 'C^θ close' should specify the space-time Hölder norm C^θ_{x,t}; the estimate follows from Proposition A.1, but this should be stated explicitly. Also, the conclusion that a nonconstant energy profile forces v to lie outside the uniform closure of smooth solutions would benefit from a one-sentence justification.
[Section 6, notation] The space cθ of little-Hölder functions is mentioned in Section 6 but is not defined there; a brief definition or reference would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's construction prescribes the energy profile and reproves the cited time-regularity lemma, so the central claims do not reduce to their inputs.

full rationale

The derivation is a convex-integration iteration in the style of Buckmaster–De Lellis–Székelyhidi–Vicol, with the energy profile e prescribed in advance rather than inferred from data. Theorem 1.1 constructs a solution realizing a given strictly positive profile; Theorem 1.2 uses the same mechanism inside a Baire-category argument to realize a profile with deliberately chosen irregularity f ∈ C^{η*} \ W^{η*}. No parameter is fitted to a subset of the target conclusion, and no prediction is an input renamed. The only author self-citation is [2] for space-time Hölder regularity of the limit, but the paper gives a complete proof of the needed statement as Proposition A.1 in the appendix, so the citation is not load-bearing. The acknowledged restrictions—working in the closure Xθ rather than all C^θ solutions, and requiring strictly positive energy profiles—are explicit limitations, not circular steps. The skeptical parameter-ordering concern about whether e_u is truly C^{η*} under the chosen ordering is a potential correctness gap in the proof as written, but it does not make the conclusion equivalent to an input by construction, so it is outside the circularity criterion. The paper's main claims are self-contained against the cited external benchmark results and do not reduce to self-referential quantities.

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

The central claim depends on the established convex integration toolkit (BDSV, Isett-Oh, Mikado flows) rather than on new fitted constants. The proof introduces auxiliary parameters a, b, α, β, η, ρ, δ, σ, all chosen by inequalities and none fitted to data. The only externally imported strong facts are the listed propositions.

assumptions (6)

standard math Mollification estimates (Proposition 2.1): ‖fδ - f‖0 ≤ δ^θ [f]_θ and related bounds.
Invoked throughout Sections 3 and 5 to control the smooth starting point and the mollified energy profile.
domain assumption Mikado flow lemma (Lemma 5.1, from [4]): for compact N⊂S_+^{3×3} there is W with prescribed Reynolds stress.
Core to the convex integration perturbation in Section 5.
domain assumption Energy regularity bound (1.2), from [8]: any C^θ solution has kinetic energy e_v ∈ C^{2θ/(1-θ)}.
Defines the target regularity class in Yθ and fixes the exponent θ* = 2θ/(1-θ).
domain assumption Cutoff functions and estimates from [1] (Lemma 5.3, Propositions 5.6 and 5.8).
Technical estimates for the Euler-Reynolds iteration are quoted rather than proved in this paper.
standard math Existence of f ∈ C^{η*} \ W^{η*} with 1/2 ≤ f ≤ 1 (for instance via [5]).
Used in Section 3.2 to force a non-improving energy profile in the Baire argument.
standard math Sobolev embedding and density of rational intervals.
Used to write the complement Yθ^c as a countable union of closed sets C_{m,n,r,s}.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Sharp energy regularity and typicality results for H\"older solutions of incompressible Euler equations." pith.science (2026). https://pith.science/paper/PT32QSUL

@misc{pith2026190803529,
  author       = {Pith},
  title        = {Pith review of: Sharp energy regularity and typicality results for H\"older solutions of incompressible Euler equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PT32QSUL}},
  note         = {Machine review of arXiv:1908.03529}
}

abstract

This paper is devoted to show a couple of typicality results for weak solutions $v\in C^\theta$ of the Euler equations, in the case $\theta<1/3$. It is known that convex integration schemes produce wild weak solutions that exhibit anomalous dissipation of the kinetic energy $e_v$. We show that those solutions are typical in the Baire category sense. From [8], it is know that the kinetic energy $e_v$ of $\theta$-H\"older continuous weak solution $v$ of the Euler equations satisfy $ e_v\in C^{\frac{2\theta}{1-\theta}}$. As a first result we prove that solutions with that behavior are a residual set in suitable complete metric space $X_\theta$, that is contained in the space of all $C^\theta$ weak solutions, whose choice is discussed at the end of the paper. More precisely we show that the set of solutions $v\in X_\theta$ with $e_v \in C^{\frac{2\theta}{1-\theta}}$ but not to $\bigcup_{p\ge 1,\varepsilon>0}W^{\frac{2\theta}{1-\theta} + \varepsilon,p}(I)$ for any open $I \subset [0,T]$, are a residual set in $X_\theta$. This, in particular, partially solves [9, Conjecture 1]. We also show that smooth solutions form a nowhere dense set in the space of all the $C^\theta$ weak solutions. The technique is the same and what really distinguishes the two cases is that in the latter there is no need to introduce a different complete metric space with respect to the natural one.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 5 canonical work pages

[2]

Regularity in time of H\"older solutions of Euler and hypodissipative Navier-Stokes equations

M. Colombo and L. De Rosa, Regularity in time of Hölder solutions of Euler and hypodiss ipative Navier-Stokes equations, preprint: https://arxiv.org/abs/1811.12870, 2018

work page Pith review arXiv 2018
[1]

Buckmaster, C

T. Buckmaster, C. De Lellis, L. Székelyhidi Jr., and V. Vi col, Onsager’s conjecture for admissible weak solutions , Comm. Pure Appl. Math. 72 (2019), no. 2, 229–274, DOI 10.1002/cpa.21781. MR3896021

doi:10.1002/cpa.21781 2019
[3]

Constantin, W

P. Constantin, W. E, and E. S. Titi, Onsager’s conjecture on the energy conservation for soluti ons of Euler’s equation , Comm. Math. Phys. 165 (1994), no. 1, 207–209. MR1298949

work page 1994
[4]

De Lellis and L

C. De Lellis and L. Székelyhidi Jr., Dissipative continuous Euler ﬂows , Invent. Math. 193 (2013), no. 2, 377–407, DOI 10.1007/s00222-012-0429-9. MR3090182

doi:10.1007/s00222-012-0429-9 2013
[5]

G. H. Hardy, Weierstrass/quotesingle.ts1s Non-Diﬀerentiable Function , Transactions of the American Mathematical Society 17 (1916), no. 3, 301, DOI 10.2307/1989005

work page doi:10.2307/1989005 1916
[6]

Isett, A proof of Onsager’s conjecture , Ann

P. Isett, A proof of Onsager’s conjecture , Ann. of Math. (2) 188 (2018), no. 3, 871–963, DOI 10.4007/annals.2018.188.3.4. MR3866888

doi:10.4007/annals.2018.188.3.4 2018
[7]

, On the Endpoint Regularity in Onsager’s Conjecture , arXiv:1706.01549 [math.AP] (2017)

arXiv 2017
[8]

, Regularity in time along the coarse scale ﬂow for the incompr essible Euler equations , arXiv:1307.0565 [math.AP]

arXiv

Show all 10 references

[9]

Isett and S.-J

P. Isett and S.-J. Oh, On the kinetic energy proﬁle of Hölder continuous Euler ﬂows , Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 3, 711–730, DOI 10.1016/j.anihpc.2016.05.002 . MR3633742

2017 doi
[10]

2, 725–804, DOI 10.1007/s00205-016-0973-3

, On Nonperiodic Euler Flows with Hölder Regularity , Archive for Rational Mechanics and Analysis 221 (2016), no. 2, 725–804, DOI 10.1007/s00205-016-0973-3. Luigi De Rosa École Polytechnique Fédérale de Lausanne, Institute of Math ematics, Station 8, CH-1015 Lausanne, Switzer-...

2016 doi

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [2]

Regularity in time of H\"older solutions of Euler and hypodissipative Navier-Stokes equations

M. Colombo and L. De Rosa, Regularity in time of Hölder solutions of Euler and hypodiss ipative Navier-Stokes equations, preprint: https://arxiv.org/abs/1811.12870, 2018

work page Pith review arXiv 2018

[2] [1]

Buckmaster, C

T. Buckmaster, C. De Lellis, L. Székelyhidi Jr., and V. Vi col, Onsager’s conjecture for admissible weak solutions , Comm. Pure Appl. Math. 72 (2019), no. 2, 229–274, DOI 10.1002/cpa.21781. MR3896021

doi:10.1002/cpa.21781 2019

[3] [3]

Constantin, W

P. Constantin, W. E, and E. S. Titi, Onsager’s conjecture on the energy conservation for soluti ons of Euler’s equation , Comm. Math. Phys. 165 (1994), no. 1, 207–209. MR1298949

work page 1994

[4] [4]

De Lellis and L

C. De Lellis and L. Székelyhidi Jr., Dissipative continuous Euler ﬂows , Invent. Math. 193 (2013), no. 2, 377–407, DOI 10.1007/s00222-012-0429-9. MR3090182

doi:10.1007/s00222-012-0429-9 2013

[5] [5]

G. H. Hardy, Weierstrass/quotesingle.ts1s Non-Diﬀerentiable Function , Transactions of the American Mathematical Society 17 (1916), no. 3, 301, DOI 10.2307/1989005

work page doi:10.2307/1989005 1916

[6] [6]

Isett, A proof of Onsager’s conjecture , Ann

P. Isett, A proof of Onsager’s conjecture , Ann. of Math. (2) 188 (2018), no. 3, 871–963, DOI 10.4007/annals.2018.188.3.4. MR3866888

doi:10.4007/annals.2018.188.3.4 2018

[7] [7]

, On the Endpoint Regularity in Onsager’s Conjecture , arXiv:1706.01549 [math.AP] (2017)

arXiv 2017

[8] [8]

, Regularity in time along the coarse scale ﬂow for the incompr essible Euler equations , arXiv:1307.0565 [math.AP]

arXiv