Absence of anomalous dissipation for vortex sheets

Helena Nussenzveig Lopes; Milton Lopes Filho; Tarek Elgindi

REVIEW 3 major objections 6 minor 1 cited by

Absence of anomalous dissipation for vortex sheets

T0 review · 3 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read Viscous energy loss vanishes for 2D vortex-sheet flows.

desk verdict A technically solid and genuinely new quantitative proof of no anomalous dissipation for vortex sheets, but the initial-time layer leans on an unstated borrowed lemma that the referee should check. read the letter →

arxiv 2504.18523 v1 pith:ZOJNZZEV submitted 2025-04-25 math.AP

Tarek Elgindi , Milton Lopes Filho , Helena Nussenzveig Lopes This is my paper · ORCID

classification math.AP MSC 35Q3035Q3176D05

keywords anomalousdissipationvortexsheetsNavier-StokesvanishingviscosityNashinequalityvorticityenergy2DEuler

topics Navier-Stokes Regularity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that families of two-dimensional Navier-Stokes solutions with vortex-sheet initial data do not exhibit anomalous dissipation: the viscous dissipation term ν∫₀ᵀ‖ω^ν‖²_{L²}dt tends to 0 as ν→0⁺. The initial vorticity may be any nonnegative bounded Radon measure plus an Lᵖ function (or L¹), the initial velocity is only in L², and an external force is allowed. The proof is quantitative and yields an explicit rate, of order |log ν|^{−1/4}, in the vortex-sheet case. A sympathetic reader should care because anomalous dissipation is a hallmark of turbulent energy loss in the inviscid limit, and this removes it for a broad class of singular planar flows.

What carries the argument

The load-bearing object is a refinement of Nash's inequality in dimension two, stated as Proposition 3.2 and extended to bounded Radon measures in Proposition 4.2. For a family of mean-free H¹ functions with uniformly bounded L¹ norm and uniformly vanishing mass in small disks, the proposition produces a convex, increasing, superquadratic function Υ such that Υ(‖f‖²_{L²}) ≤ ‖∇f‖²_{L²}; when f is a nonnegative measure plus an Lᵖ function, Υ can be chosen so that its inverse-square-root behaves like x|log x|^{−1/4} at infinity. Applied to the vorticity ω^ν(·,t) at each time, this inequality converts the known global bound on ‖∇ω^ν‖²_{L²} into a lower bound on ν²∫Υ(‖ω^ν‖²)dt, and Jensen's inequality turns that into the claimed vanishing rate for the dissipation.

What would settle it

A decisive observation would be a family satisfying hypotheses H(a)–H(b) and H2(a)–H2(b) for which limsup_{ν→0⁺} ν∫₀ᵀ‖ω^ν‖²_{L²}dt > 0, or alternatively a demonstration that the L² right-continuity lemma fails under the stated hypotheses. The Section 7 Dirac-type example shows how the conclusion fails when the no-Diracs and strong-initial-data conditions are dropped, but it does not violate the hypotheses of the theorems, so it does not falsify them.

Watch

Extended reading notes

Core claim

The central claim is Theorem 6.4 together with Remark 6.6: if the initial vorticities decompose as μ^ν + w^ν with μ^ν ≥ 0 bounded in BM, w^ν bounded in Lᵖ for some p > 1, and the curls of the forcings are bounded in L¹(0,T;Lᵖ), then every physically realizable weak limit satisfies limsup_{ν→0⁺} ν∫₀ᵀ‖ω^ν(·,t)‖²_{L²}dt = 0, with the dissipation on any interval [δ,T] bounded by a constant times |log ν|^{−1/4}. A more general theorem (Theorem 5.2) proves the same vanishing conclusion for any vanishing-viscosity family satisfying hypotheses H(a)–H(c) plus the no-Diracs condition that vorticity mass in small disks tends to zero uniformly. The authors also show the method extends to a time-averaged version of the no-Diracs condition and yields the same rate for the singular part. This is a statement about absence of anomalous dissipation only; it does not assert strong convergence of the velocity approximations.

Load-bearing premise

The argument rests on a lemma borrowed from the companion preprint [11] that the limiting Euler solution is right-continuous in L² at t=0; if that lemma requires hypotheses stronger than those verified here, the treatment of the initial time layer [0,δ] would have a gap.

Editorial extensions

If this is right

Dissipation vanishes for vortex-sheet initial data: ν∫₀ᵀ‖ω^ν‖²_{L²}dt → 0 as ν→0⁺, even though the velocity family need not converge strongly in L².
The rate |log ν|^{−1/4} quantifies how quickly the dissipation disappears; the paper does not claim this rate is optimal.
Forced flows are covered, so the result applies beyond the unforced Navier-Stokes setting.
Absence of anomalous dissipation is compatible with the limiting weak solution carrying an energy defect; it is strictly weaker than strong convergence.
The Section 7 example shows the conclusion can fail when both the no-Diracs and strong-initial-data conditions are violated, so the hypotheses are not vacuous.

Reading between the lines

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

The logarithmic rate suggests the mechanism is robust: any singularity class whose small-scale mass decays at least logarithmically may yield a vanishing rate, so the method could extend to other critical Besov or Morrey-type initial data.
The time-averaged no-Diracs extension in Section 7 implies the result should be stable under oscillations in time of the vortex sheet, not just spatial concentration.
A natural test is whether the |log ν|^{−1/4} rate is sharp; constructing an example that saturates the bound would reveal whether the refined Nash inequality is lossless.
Combined with the equivalence proved in the companion preprint [11], absence of anomalous dissipation is necessary but not sufficient for energy balance of the inviscid limit; the missing ingredient is strong L² convergence, which this method does not attempt to provide.

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

Here's my take: this is a real result, with a genuinely new technical tool, but the qualitative conclusion was already out there. De Rosa and Park proved no anomalous dissipation for vortex sheets; Elgindi et al. prove it again, with forcing, more flexible approximation, and an explicit |log nu|^{-1/4} rate for the measure+L^p case. They are upfront about the overlap. The new refined Nash inequality (Prop 3.2 and its measure-valued version Prop 4.2) is the real contribution, and it looks correct to me.

The proof is mostly careful and readable. Lemmas 2.1-2.2 and the Nash refinements are worked out in detail. The structure of Theorem 5.2 is sound: split the dissipation into [delta,T] and [0,delta]; on the body, the refined inequality plus Jensen gives the decay; on the initial layer, the energy equation and weak lower-semicontinuity reduce things to the energy gap of the limit.

The soft spot is exactly the initial layer. Equation (5.17) leaves a term involving ||u0||^2 - ||u(delta)||^2, and the proof lets delta go to 0 using a cited lemma [11, Lemma 3.5] that the inviscid weak solution is right-continuous in L^2 at t=0. The lemma is not stated, and its hypotheses are not checked in the text. This is not a manufactured flaw: the claim is load-bearing. Without right-continuity, the initial layer might not vanish, and then the theorem does not close. I suspect the lemma is true under the given hypotheses, but I could not verify it from this paper. The authors should either prove it or state it precisely and verify its hypotheses. The same goes for the use of [11, Prop 4.5] in Theorem 6.1, and Corollary 6.7 is omitted as 'standard'.

Minor: the abstract says 'provides an explicit estimate for the dissipation'; the explicit rate appears only in Theorem 6.4. In Theorem 5.2 the rate is non-explicit. Slightly soften that.

This is for people working on vanishing viscosity and 2D turbulence; the inequality itself is reusable. Bottom line: it deserves a serious referee. The analytic core is new and the conclusion is important, even if not first. I would accept for review, with a referee asked to push for a self-contained treatment of the initial-time layer and the borrowed lemmas.

Referee Report

3 major / 6 minor

Summary. The paper proves absence of anomalous dissipation for families of 2D Navier-Stokes solutions on the torus whose initial vorticities are nonnegative bounded Radon measures plus an L^p function, with L^2 initial velocity, allowing external forcing. The main analytic tool is a new refinement of Nash's inequality: for families of mean-free H^1 functions with uniform L^1 bound and uniformly vanishing mass in small balls, a convex increasing superquadratic function Y satisfies Y(||f||_{L^2}^2) <= ||∇f||_{L^2}^2. A measure-valued version gives the explicit growth Y(x) ∼ x|log x|^{-1/4}. These estimates are fed into the vorticity equation to show limsup_{ν→0+} ν∫_0^T ||ω^ν||_{L^2}^2 dt = 0 under a no-Diracs condition on the vorticity family, and the hypotheses are then verified for vortex-sheet initial data in BM^+ + L^p, p>1, with an explicit |log ν|^{-1/4} rate; the L^1 case is stated as Corollary 6.7. The paper also discusses an extension to time-averaged no-Diracs conditions and a counterexample illustrating the role of the hypotheses.

Significance. If the proof is completed as indicated, this is a substantial contribution: it extends the recent no-anomalous-dissipation result of De Rosa and Park to forced flows, allows measure-plus-L^p initial vorticities, and provides a quantitative vanishing rate. The refined Nash inequality in Propositions 3.2 and 4.2 is a clean and potentially reusable real-analysis result independent of the fluid application. The paper is honest about the role of the no-Diracs condition and gives a concrete example showing that both strong convergence of initial data and the no-Diracs condition are needed. The main weakness is that several load-bearing statements are cited from the unpublished preprint [11] without statement or proof, in one case closing the initial-time layer of the central theorem.

major comments (3)

[§5, proof of Theorem 5.2, Eq. (5.17)] The concern raised about the initial-time layer is justified. After deriving (5.17), the paper closes the limit δ→0+ by invoking [11, Lemma 3.5] for L^2 right-continuity of the limiting Euler solution at t=0. This lemma is load-bearing: without it, the term (1/2)(||u_0||^2 - ||u(δ)||^2) in (5.17) need not vanish, and no other argument in the paper controls ∫_0^δ ||ω^ν||^2 dt. Since [11] is a preprint by two of the present authors and the lemma is neither stated nor proved here, the main theorem is conditional on an unverified external assumption. The authors should state the lemma with precise hypotheses and either prove it in an appendix or replace it with a published reference that covers exactly the present hypotheses.
[§6, Corollary 6.7] The abstract advertises the main result for 'an arbitrary non-negative measure plus an integrable function' as initial vorticity, which is exactly the content of Corollary 6.7 with H3(a)-H3(b). However, the proof is omitted as 'standard'. Since this corollary is one of the paper's advertised results and depends on the same initial-layer argument as Theorem 5.2, the omission is a gap in the manuscript's central claim. The proof should be included, or the abstract and corollary should be explicitly downgraded to a conjecture/sketch.
[§6, proof of Theorem 6.1, Eq. (6.1)] The verification of the no-Diracs condition (5.3) relies on the maximal-function estimate (6.1), quoted from [11, Proposition 4.5] without proof. This proposition is used to propagate uniform integrability of the initial and forcing vorticities to all times, so it is essential for Theorem 6.1 to imply H(c) and (5.3). Like Lemma 3.5, this should be stated explicitly and either proved or given a fully verifiable published reference, since [11] is not yet part of the public peer-reviewed record.

minor comments (6)

[§5, text after (5.17)] The sentence citing [11, Lemma 3.5] gives no indication of the lemma's hypotheses; even a brief statement of the exact assumptions would help the reader see that they match H(a)-H(c) and (5.3).
[§4, Proposition 4.1 proof] In the line after (4.4), the expression '||µ||_{H^{-1}}| + ||w||_{L^p}' contains a stray vertical bar after the H^{-1} norm; this is likely a typographical error.
[§3-§4, notation] The notation '4√log' in (4.2), (4.6), (4.7) is ambiguous; writing (log x)^{1/4} or \sqrt[4]{\log x} would make the estimates easier to read.
[§2, Lemma 2.2] The statement uses N ∈ R^+ but then conflates N with the integer ⌈N⌉ in the proof; a remark clarifying that all estimates are unchanged up to constants would remove the ambiguity.
[§7, extension discussion] The claimed generalizations of Propositions 3.1, 3.2, 4.1, and 4.2 to time-averaged conditions (B') and (B'') are only sketched. Since these are presented as extensions rather than as main theorems, a sketch may be acceptable, but the definitions of the analogous η and Φ functions should at least be written down.
[§7, example] In the example, the notation '~Cδ0' appears instead of a properly typeset multiple of a Dirac delta; this should be corrected.

Circularity Check

2 steps flagged · score 4.0 of 10

The core Nash-refinement derivation is independent, but load-bearing steps in The proof of Theorems 5.2 and 6.1 are justified by two unproved lemmas from the same authors' preprint [11].

self citation load bearing [Section 5, proof of Theorem 5.2, after Eq. (5.17)]
"Now, it was established in [ 11, Lemma 3.5] that, under the current hypotheses, the inviscid solution is right-continuous in L2 at t = 0, that is, lim t→0+ ‖u(t) − u0‖2 L2 = 0, Hence the right-hand-side of ( 5.17) is vanishingly small as δ → 0+."

The initial-layer estimate (5.17) gives limsup_{ν→0+} ν∫_0^δ ‖ω^ν‖² dt ≤ 1/2(‖u0‖²−‖u(δ)‖²)+C√δ, and sending δ→0+ requires the L² right-continuity asserted only by the citation to [11, Lemma 3.5]. That preprint is co-authored by two of the present authors, and the lemma is neither stated nor proved here, nor is any check given that its hypotheses coincide with H(a)-H(c)+(5.3). This makes a load-bearing step of the central theorem reduce to an unverified self-citation rather than to the paper's own hypotheses; if the lemma requires extra assumptions, Theorem 5.2 is not closed.
self citation load bearing [Section 6, proof of Theorem 6.1, Eq. (6.1)]
"The following estimate was established in [ 11, Proposition 4.5]: Ms(ων(t)) ≤ Ms(ων 0 ) + ˆT 0 Ms(curl F ν(τ )) d τ. (6.1) We use (6.1) to propagate the uniform integrability of ων 0 and curl F ν(τ ), which follow from H1(a) and H1(b). We conclude that Ms(ων(t)) → 0 as s → 0, uniformly in t ∈ [0, T ] and ν > 0, and, consequently, ( 5.3) holds true."

Theorem 6.1 reduces the vortex-sheet case to Theorem 5.2 by verifying the no-Diracs condition (5.3), and the verification consists entirely of the maximal-function estimate (6.1) imported from [11], a preprint co-authored by two of the present authors. The estimate is not derived in this paper, and it is the only mechanism by which the data-only hypotheses H1/H2 imply (5.3); without it, the application of Theorem 5.2 fails. This is a second load-bearing self-citation used to bridge from initial-data assumptions to the main theorem.

full rationale

The central derivation is not circular in the definitional sense: the Nash-refinement inequalities (Propositions 3.1, 3.2, 4.1, and 4.2) are proved in the paper, the body estimate (5.15) follows from them by Jensen's inequality with no fitted parameters, and the rate |log ν|^{-1/4} in Remark 6.6 is read off the constructed function Φ rather than imposed by data. The hypothesis (5.3) is a concentration/no-Diracs condition, not the conclusion ν∫‖ω^ν‖² dt→0, and the paper's own Section 7 example shows the two are not interchangeable. However, the proof is not fully self-contained: it imports two lemmas from the same authors' preprint [11]. Lemma 3.5 (L² right-continuity at t=0) is load-bearing for the initial-layer estimate (5.17) in Theorem 5.2, and Proposition 4.5 (maximal-function propagation) is the sole route from data-only assumptions to (5.3) in Theorem 6.1; both are cited but not proved or checked against the present hypotheses. Corollary 6.7 is also stated with its proof omitted ('The proof is a standard combination of ideas from the proofs of Theorem 6.1 and Theorem 6.4, so we omit it.'). Because these are supporting self-citations rather than an equation that is its own input, the appropriate score is a moderate 4 rather than a high circularity score.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new free parameters and no invented entities. Its central claim rests on standard PDE existence theory, several cited lemmas from prior work (some self-cited from [11]), and a new analytic inequality derived in the paper itself.

assumptions (7)

domain assumption Leray-Hopf existence, uniqueness and energy equality for 2D Navier-Stokes with L^2 initial data and L^2 forcing
Used throughout Section 5; standard, but the energy identity (5.4) relies on it.
domain assumption Delort's compensated-compactness theorem for vortex sheets of distinguished sign, and its adaptation used to identify the weak limit as an Euler solution
Used after (5.16) to pass to the inviscid limit; cited to [4,20].
domain assumption Schochet's small-scale mass estimate for nonnegative measures in H^{-1}: integral over a ball of radius rho is bounded by C||mu||_{H^{-1}} |log rho|^{-1/2}
Used in Proposition 4.1, equation (4.1), cited to [20].
domain assumption Right-continuity in L^2 at t=0 of the limiting weak Euler solution, [11, Lemma 3.5]
Used at the end of Theorem 5.2 to make the initial-layer dissipation vanish; load-bearing and self-cited.
domain assumption Propagation of rearrangement-invariant maximal functions for vorticity, [11, Proposition 4.5], namely M_s(omega^nu(t)) <= M_s(omega^nu_0) + integral M_s(curl F^nu) dt
Used in Theorem 6.1 to verify (5.3) from weak compactness.
domain assumption Parabolic maximum principle and sign preservation for the linear vorticity transport equation with drift u^nu
Used in Theorem 6.4 to prove mu^nu(t) >= 0 and L^1 bounds.
standard math Standard real analysis: Riesz-Thorin, Young, Plancherel, Poincare, Jensen inequalities
Used in Sections 2-3 for the convolution estimates and the Nash refinement.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Absence of anomalous dissipation for vortex sheets." pith.science (2026). https://pith.science/paper/ZOJNZZEV

@misc{pith2026250418523,
  author       = {Pith},
  title        = {Pith review of: Absence of anomalous dissipation for vortex sheets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZOJNZZEV}},
  note         = {Machine review of arXiv:2504.18523}
}

read the original abstract

A family of solutions of the incompressible Navier-Stokes equations is said to present anomalous dissipation if energy dissipation due to viscosity does not vanish in the limit of small viscosity. In this article we present a proof of absence of anomalous dissipation for 2D flows on the torus, with an arbitrary non-negative measure plus an integrable function as initial vorticity and square-integrable initial velocity. Our result applies to flows with forcing and provides an explicit estimate for the dissipation at small viscosity. The proof relies on a new refinement of a classical inequality due to J. Nash.

Discussion (0). Continue with ORCID 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

Global Existence, Hamiltonian Conservation and Vanishing Viscosity for the Surface Quasi-Geostrophic Equation
math.AP 2025-09 accept novelty 7.0 of 10

For any L^{4/3}_x initial datum there exists a global weak solution of SQG conserving the H^{-1/2}_x Hamiltonian, obtained via a vanishing-viscosity limit with no anomalous dissipation.

Reference graph

Works this paper leans on

21 extracted references · 20 canonical work pages · cited by 1 Pith paper

[11]

F. Jin, S. Lanthaler, M. C. L. Filho, and H. J. N. Lopes, Sharp conditions for energy balance in two-dimensional incompressible ideal ﬂo w with external force (2024). ↑2, 14, 16, 18, 22

work page 2024
[1]

Bouin, J

E. Bouin, J. Dolbeault, and C. Schmeiser, A variational proof of Nash’s in- equality, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Linc ei, Mat. Appl. 31 (2020), no. 1, 211–223 (English). ↑ 6

work page 2020
[2]

Bru` e, M

E. Bru` e, M. Colombo, and A. Kumar, Flexibility of two-dimensional euler ﬂows with integrable vorticity , 2024. ↑13

work page 2024
[3]

Nussenzveig Lopes, and R Shvydko y, Energy conservation in two-dimensional incompressible ideal ﬂui ds, Communications in Mathematical Physics 348 (2016), no

A Cheskidov, M Lopes Filho, H. Nussenzveig Lopes, and R Shvydko y, Energy conservation in two-dimensional incompressible ideal ﬂui ds, Communications in Mathematical Physics 348 (2016), no. 1, 129–143. ↑13

work page 2016
[4]

Delort, Existence of vortex sheets in dimension two , J

J.-M. Delort, Existence of vortex sheets in dimension two , J. Am. Math. Soc. 4 (1991), no. 3, 553–586 (French). ↑2, 9, 13, 16, 19, 21

work page 1991
[5]

R. J. DiPerna and A. Majda, Reduced Hausdorﬀ dimension and concentration- cancellation for two- dimensional incompressible ﬂow , J. Am. Math. Soc. 1 (1988), no. 1, 59–95 (English). ↑12

work page 1988
[6]

R. J. Diperna and A. J. Majda, Concentrations in regularizations for 2 − D incompressible ﬂow , Commun. Pure Appl. Math. 40 (1987), no. 3, 301–345 (English). ↑12

work page 1987
[7]

R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solu- tions of the incompressible ﬂuid equations , Commun. Math. Phys. 108 (1987), 667–689 (English). ↑12

work page 1987

Show all 21 references

[8]

G. B. Folland, Real analysis. Modern techniques and their applications. , 2nd ed., Pure Appl. Math., Wiley-Intersci. Ser. Texts Monogr. Tracts, New York, NY: Wiley, 1999 (English). ↑4

1999
[9]

Frisch, Turbulence: The legacy of A

U. Frisch, Turbulence: The legacy of A. N. Kolmogorov , Cambridge University Press, 1995. ↑2

1995
[10]

Iftimie, M

D. Iftimie, M. C. Lopes Filho, and H. J. Nussenzveig Lopes, Weak vortic- ity formulation of the incompressible 2D Euler equations in bounded domains, Comm. Partial Diﬀerential Equations 45 (2020), no. 2, 109–145. MR4045802 ↑16

2020
[12]

Lanthaler, S

S. Lanthaler, S. Mishra, and C. Par´ es-Pulido, On the conservation of energy in two-dimensional incompressible ﬂows , Nonlinearity 34 (2021), no. 2, 1084, available at 2001.06195. ↑13

2021 arXiv
[13]

M. C. Lopes Filho, H. J. Nussenzveig Lopes, and Z. Xin, Existence of vortex sheets with reﬂection symmetry in two space dimensions , Arch. Ration. Mech. Anal. 158 (2001), no. 3, 235–257. MR1842346 ↑21

2001
[14]

Lopes Filho and H

M. Lopes Filho and H. Nussenzveig Lopes, Energy balance for forced two- dimensional incompressible ideal ﬂuid ﬂow , Philosophical Transactions of the Royal Society A 380 (2022), no. 2219, 20210095. ↑13 24 ELGINDI, LOPES FILHO, AND NUSSENZVEIG LOPES

2022
[15]

M. C. Lopes Filho, H. J. Nussenzveig Lopes, and E. Tadmor, Approximate solutions of the incompressible Euler equations with no con centrations, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire17 (2000), no. 3, 371–412. MR1771138 ↑ 2

2000
[16]

A. J. Majda, Remarks on weak solutions for vortex sheets with a distingui shed sign, Indiana Univ. Math. J. 42 (1993), no. 3, 921–939 (English). ↑2, 19

1993
[17]

J. F. Nash, Continuity of solutions of parabolic and elliptic equation s, Am. J. Math. 80 (1958), 931–954 (English). ↑6

1958
[18]

L. D. Rosa and J. Park, No anomalous dissipation in two-dimensional incom- pressible ﬂuids (2024), available at 2403.04668. ↑3, 21

2024 arXiv
[19]

Rudin, Principles of mathematical analysis

W. Rudin, Principles of mathematical analysis. 3rd ed , 1976 (English). ↑5

1976
[20]

Schochet, The weak vorticity formulation of the 2-D Euler equations an d concentration- cancellation , Commun

S. Schochet, The weak vorticity formulation of the 2-D Euler equations an d concentration- cancellation , Commun. Partial Diﬀer. Equations 20 (1995), no. 5-6, 1077–1104 (English). ↑9, 16, 19, 21

1995
[21]

Vecchi and S

I. Vecchi and S. Wu, On L1-vorticity for 2-D incompressible ﬂow , Manuscr. Math. 78 (1993), no. 4, 403–412 (English). ↑2, 12, 18 1 Department of Mathematics, Duke University, Durham, NC 277 08- 0320 – USA Email address : tarek.elgindi@duke.edu 2 Instituto de Matem ´atica, Univ...

1993

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [11]

F. Jin, S. Lanthaler, M. C. L. Filho, and H. J. N. Lopes, Sharp conditions for energy balance in two-dimensional incompressible ideal ﬂo w with external force (2024). ↑2, 14, 16, 18, 22

work page 2024

[2] [1]

Bouin, J

E. Bouin, J. Dolbeault, and C. Schmeiser, A variational proof of Nash’s in- equality, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Linc ei, Mat. Appl. 31 (2020), no. 1, 211–223 (English). ↑ 6

work page 2020

[3] [2]

Bru` e, M

E. Bru` e, M. Colombo, and A. Kumar, Flexibility of two-dimensional euler ﬂows with integrable vorticity , 2024. ↑13

work page 2024

[4] [3]

Nussenzveig Lopes, and R Shvydko y, Energy conservation in two-dimensional incompressible ideal ﬂui ds, Communications in Mathematical Physics 348 (2016), no

A Cheskidov, M Lopes Filho, H. Nussenzveig Lopes, and R Shvydko y, Energy conservation in two-dimensional incompressible ideal ﬂui ds, Communications in Mathematical Physics 348 (2016), no. 1, 129–143. ↑13

work page 2016

[5] [4]

Delort, Existence of vortex sheets in dimension two , J

J.-M. Delort, Existence of vortex sheets in dimension two , J. Am. Math. Soc. 4 (1991), no. 3, 553–586 (French). ↑2, 9, 13, 16, 19, 21

work page 1991

[6] [5]

R. J. DiPerna and A. Majda, Reduced Hausdorﬀ dimension and concentration- cancellation for two- dimensional incompressible ﬂow , J. Am. Math. Soc. 1 (1988), no. 1, 59–95 (English). ↑12

work page 1988

[7] [6]

R. J. Diperna and A. J. Majda, Concentrations in regularizations for 2 − D incompressible ﬂow , Commun. Pure Appl. Math. 40 (1987), no. 3, 301–345 (English). ↑12

work page 1987

[8] [7]

R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solu- tions of the incompressible ﬂuid equations , Commun. Math. Phys. 108 (1987), 667–689 (English). ↑12

work page 1987