An Onsager-type Theorem for General 2D Active Scalar Equations

Xuanxuan Zhao

REVIEW 3 major objections 4 minor 1 cited by

An Onsager-type Theorem for General 2D Active Scalar Equations

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For every odd active scalar equation in two dimensions, the paper constructs non-conservative Hölder solutions up to the sharp Onsager threshold.

desk verdict Plausibly correct and important result for the flexible part of the Onsager-type conjecture for general odd 2D active scalars, but the proof's key trilinear estimate is not fully verified and the stated regularity is overstated. read the letter →

arxiv 2412.11094 v2 pith:MV4IINOS submitted 2024-12-15 math.AP

Xuanxuan Zhao This is my paper · ORCID

classification math.AP MSC 35Q3535D3076B0342B15

keywords OnsagerconjectureactivescalarequationsconvexintegrationNewton-NashiterationHamiltonianconservationtrilinearFouriermultiplierssurfacequasi-geostrophicweaksolutions

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 aims to establish the flexible half of the generalized Onsager conjecture for all two-dimensional odd active scalar equations: $\partial_t\theta + u\cdot\nabla\theta = 0$ with $u = T[\theta]$ a divergence-free Fourier multiplier whose odd, $\delta$-homogeneous symbol satisfies $-1 \le \delta \le 0$. The main theorem asserts that for every $(1+\delta)/2 \le \gamma < 1+2\delta/3$ there exist non-trivial, temporally compact-supported weak solutions with $\Lambda^{-1}\theta \in C^\gamma$ that violate Hamiltonian conservation. This is exactly the regularity window left open by the rigidity side in the reference cited as [23], so the result locates the phase transition for Hamiltonian conservation at the exponent $1+2\delta/3$. The proof unifies the sharp flexible results for the 2D Euler vorticity equation and the SQG equation, and the appendix treats even multipliers as well.

What carries the argument

The argument is carried by a Newton-Nash convex integration scheme lifted to the potential velocity $v = (-\Delta)^{-1}\nabla^\perp\theta$, whose momentum system has velocity $u = T_1[v]$ with an even multiplier of order $1+\delta$; this extra derivative compensates for the negative homogeneity of $m$. The scheme rests on three custom pieces: a classification of $m$ into two or three independent trace-free tensors $\xi \otimes \nabla\bar{m}(\xi)$, which yields an $m$-dependent anti-divergence operator and the algebraic Lemma 3.2 that decomposes the Reynolds stress into elementary tensors; sharp estimates for the trilinear Fourier multiplier commutators $S^0$, $S^i$ and $S^{i,j}$ in Lemma 4.5, which control material derivatives without derivative loss; and a multi-step Newton iteration with temporally disjoint profiles whose number of steps $\Gamma$ is chosen so that gluing errors fall below the Nash error.

What would settle it

Evaluate the symbol $M_{s,i}(\xi,\eta,\zeta)$ of Lemma 4.5 on frequency triads with $|\xi| \ll |\eta| \approx |\zeta|$ and check whether the claimed bound (4.12) holds; if one triad violates it, the material-derivative estimate on the Newton stress error in Proposition 5.6 breaks, so constructing such a counterexample would disprove the convergence proof.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 1.3: for any odd, divergence-free, $\delta$-homogeneous Fourier multiplier $m$ smooth away from the origin with $-1 \le \delta \le 0$, and for any exponent $(1+\delta)/2 \le \gamma < 1+2\delta/3$, the active scalar equation admits non-trivial weak solutions $\theta$ with compact temporal support and $\Lambda^{-1}\theta \in C^\gamma(\mathbb{R} \times \mathbb{T}^2)$. These solutions fail to conserve the Hamiltonian, while the companion rigidity result [23] guarantees conservation above that threshold; together the two statements identify $1+2\delta/3$ as the sharp critical exponent. In the appendix, an analogous construction for even multipliers in two and three dimensions yields non-trivial Hölder solutions with exponent below $1/3$, matching the known energy-conservation threshold.

Load-bearing premise

The iteration closes only if the sharp trilinear Fourier multiplier estimate in Lemma 4.5, for the commutators $S^0[u,\varphi,\psi]$, $S^i[u,\varphi,\psi]$ and $S^{i,j}[u,\varphi,\psi]$, is valid in every frequency arrangement for the full range $-1 \le \delta \le 0$, and the proof of that lemma is where several steps are only summarized.

Editorial extensions

If this is right

Hamiltonian conservation for general 2D odd active scalars now has a sharp threshold: weak solutions conserve the Hamiltonian above $C^{1+2\delta/3}$ and can violate it strictly below.
The same iteration applies uniformly to the mSQG family, recovering the sharp flexible results for 2D Euler in vorticity form and for SQG as endpoints $\delta = -1$ and $\delta = 0$.
The non-conservative solutions are compactly supported in time, so the theorem also implies failure of uniqueness and of compactness of solution sets at those regularities.
For even multipliers, the appendix gives non-trivial $C^\alpha$ solutions with $\alpha < 1/3$, matching the energy-conservation threshold known from the positive side.

Reading between the lines

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

Beyond the paper, the tensor hierarchy sketched in Section 1.3 suggests the same construction should extend to $\delta < -1$ by iterating potential fields of higher order; a testable project is to formulate the multilinear estimates those higher systems require.
The multiplier classification in Section 3.1 reveals that the exceptional symbols blocking the algebraic lemma are precisely the higher-order polynomial-like cases and the degenerate stationary case $m \propto i\xi^\perp$, which hints that any future treatment of $\delta > 0$ must first handle these exceptions.
A numerical check of the symbol $M_{s,i}(\xi,\eta,\zeta)$ on frequency triads with $|\xi| \ll |\eta| \approx |\zeta|$ would test whether the cancellations asserted in Lemma 4.5 hold as stated.

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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

Short version: this is a serious paper on the right problem. The main theorem – existence of non-conservative weak solutions with Λ^{-1}θ ∈ C^γ for every γ < 1 + 2δ/3, matching the Isett–Ma rigidity threshold – is a genuine advance. Prior sharp flexible constructions only covered Euler (δ=−1) and SQG (δ=0); the full range −1 ≤ δ ≤ 0 was open. If correct, this closes the flexible part of the generalized Onsager conjecture for odd active scalars. That is worth a referee's time.

The architecture is coherent: working at the level of the potential v = −∇⊥(−Δ)^{-1}θ buys one degree of homogeneity, the adapted Newton–Nash iteration from Giri–Radu is appropriate, and the algebraic lemma for decomposing stresses into ξ ⊗ ∇m̄(ξ) with a multiplier-dependent anti-divergence operator is a real idea. The trilinear Fourier multiplier estimates are the technical heart, and the paper is honest about where its novelty lies.

Now the soft spots, in proportion. The proof of Lemma 4.5 is not complete. That lemma is load-bearing: Proposition 5.6's Newton closure depends on estimate (4.12), and the decisive frequency regime (Step 3.3) is dispatched with 'some tedious calculations' and 'one can verify'. The stress test is concrete: if any L1 kernel bound there carries an extra factor λ_{q+1}/λ_q, the closure fails. I am not asserting it does fail – the frequency configurations look plausible, and the statements of the estimates are consistent with the claimed result – but a referee must fill in those details before the paper can be accepted.

Two smaller issues. Theorem 1.3 states joint C^γ(R×T^2) regularity, but the proof constructs C^0_t C^γ_x; the theorem statement should be corrected. And Appendix D, on even multipliers, is explicitly a sketch; it is a secondary result and clearly marked as such, so I would not hold the paper to the same standard there.

Bottom line: this deserves a serious referee. The main theorem is likely true, the framework is right, and the gaps are specific and apparently fillable. I would send it to peer review and ask the referee to focus on Lemma 4.5 and the regularity statement.

Referee Report

3 major / 4 minor

Summary. The paper claims to prove the flexible part of a generalized Onsager conjecture for general 2D odd active scalar equations. For a Fourier multiplier symbol m that is odd, divergence-free, homogeneous of order δ ∈ [-1,0], and smooth away from the origin, the main theorem asserts that for every (1+δ)/2 ≤ γ < 1+2δ/3 there exist non-trivial weak solutions with compact temporal support such that Λ^{-1}θ ∈ C^γ(R_t × T^2_x), and these solutions fail to conserve the Hamiltonian. The proof works at the level of the potential field v = -∇^⊥ θ, reformulating the equation as an ASM-Reynolds system, and uses a Newton-Nash convex integration scheme following Giri-Radu. The main technical novelty is a set of bilinear and trilinear Fourier multiplier estimates, in particular Lemma 4.5, together with an algebraic lemma that provides an anti-divergence operator adapted to the general multiplier.

Significance. If the proof is completed, this would resolve the flexible part of the Onsager-type conjecture for the entire class of odd active scalar equations with homogeneous symbols of order δ ∈ [-1,0], unifying the previously known cases of 2D Euler (δ=-1) and SQG (δ=0), and matching the rigidity results of Isett-Ma. The paper introduces a genuinely new algebraic decomposition for the stress in the absence of symmetry of the tensor ξ ⊗ ∇ m̄(ξ), and the trilinear Fourier multiplier framework appears to be novel. These are potentially significant contributions to the convex integration literature. However, the central technical lemma is not fully proved in the submitted text, and the stated regularity of the solutions exceeds what the proof establishes.

major comments (3)

[Section 4, Lemma 4.5, Step 3.3] The proof of Lemma 4.5 asserts the L1 kernel bound ||F^{-1}(M^{...}_{s,i,k,ℓ})||_{L1} ≲ 1 in Step 3.3 by 'some tedious calculations' without displaying the derivative estimates. These bounds are load-bearing: Proposition 5.6 invokes (4.12) for the E4 terms, and a loss of a single power of 2^{k-k1} or 2^{k2-k} in this regime would break the Newton-step closure. Please provide the explicit derivative bounds for the symbols M^3,...,M^6 and the resulting kernel estimates in Steps 3.1-3.3 (and the analogous claims in Steps 4-6), with all powers of the frequency scales recorded.
[Theorem 1.3 and Section 2.3] Theorem 1.3 states Λ^{-1}θ ∈ C^γ(R_t × T^2_x), but the proof in Section 2.3 only establishes that {v_q} is Cauchy in C^0_t C^γ_x and that R_q → 0 in that space. No estimates controlling the time Hölder seminorm are given. The theorem should be restated as C^0_t C^γ_x, or the missing time-regularity argument should be provided.
[Lemma 4.4 and Lemmas 6.8-6.9] Lemma 4.4 is stated with 'the proof is almost the same ... and thus omitted', and Lemmas 6.8 and 6.9 similarly omit details. These lemmas feed directly into the estimates of the bilinear and trilinear multiplier bounds used in Propositions 5.6 and 6.14. Given that Lemma 4.3's proof is lengthy and the operators in Lemma 4.4 have a different structure, the omission leaves a gap in the verification of (4.6)-(4.7). Please include the proofs or a precise reduction to Lemma 4.3.

minor comments (4)

[Abstract and Theorem 1.3] The abstract states 'θ ∈ C^0_t C_x^{2δ/3-}', but Theorem 1.3 concerns Λ^{-1}θ ∈ C^γ; the relationship between these two regularities should be clarified and the apparent mismatch resolved.
[Appendix D] Theorem D.1 is stated for even multipliers, but the proof is only a sketch that ends with 'we just conclude the main results here without providing details'. If this theorem is part of the paper's claims, it requires a complete proof or should be explicitly labeled as a conjecture.
[Throughout] There are numerous typos and formatting artifacts, including 'comlpex', 'ﬁled', 'Lagraingian', and a missing summation sign in equation (4.16) where 'usBs' appears without denoting the sum over s.
[Section 5.3, equation (5.16)] The transport equation for ψ_{k,n+1} is solved with initial data at t=t_k, but the existence of a unique solution on the support of ˜χ_k should be justified using the flow estimates of Lemma 5.2; this is standard, but it should be stated explicitly for completeness.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the flexible construction is forward and self-contained, and the rigid bound is imported from external work.

full rationale

The paper's main claim is a convex-integration/Newton-Nash construction of weak solutions, not a fitted prediction. The construction parameters β, b, α, M, and a are chosen to close the inductive estimates (2.10)-(2.11) and the resulting regularity is read off from the parameter β; the target exponent γ is not inserted as an input and then recovered by definition. The sharp rigidity threshold 1+2δ/3 is cited from Isett-Ma [23], an external result by other authors, and the flexible part does not rely on that citation for its construction. Prior works [17], [10], [21], [22] are cited as frameworks and techniques, not as self-citations carrying the argument. The algebraic lemma and anti-divergence operator are proved inside the paper, and the trilinear Fourier multiplier estimates in Lemma 4.5 are internal technical tools; even if their proof is incomplete or insufficiently detailed, that is a correctness or rigor concern rather than circularity. No step in the derivation reduces by construction to its own inputs, and no fitted parameter is renamed as a prediction. The reviewer's concern about omitted details in Lemma 4.5 and about the joint-in-time regularity statement is a correctness risk, not a circularity, and does not raise the circularity score.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

No physical entities are postulated. The listed free parameters are standard convex integration constants with no empirical content. The axioms include the defining class assumptions, the external rigidity result, the borrowed Newton-Nash framework, standard harmonic analysis tools, and the paper's own incompletely proved trilinear estimate.

free parameters (7)

beta = chosen in ((1+delta)/2, 1+2delta/3), close to the upper endpoint
Controls delta_q = lambda_q^{-2beta} and hence the Holder exponent of the constructed solution; must satisfy condition (2.10). No empirical content.
b = b > 1, chosen sufficiently close to 1
Frequency growth parameter in lambda_q = a^{b^q}; chosen so that (2.10) is compatible with beta near 1+2delta/3.
a = a > a0, sufficiently large
Starting frequency parameter; all inductive inequalities are closed by taking a large enough.
alpha = 0 < alpha < alpha0, sufficiently small
Small Holder slack used in the Fourier multiplier estimates and mollification; chosen small to close the final estimates.
M = M > M0, sufficiently large
Amplitude constant in the inductive bounds (2.6)-(2.8); chosen large enough to absorb constants.
mu_{q+1} = delta_{q+1}^{1/2} lambda_{q+1}^{1+2delta/3} lambda_q^{1+delta/3} lambda_{q+1}^{4alpha}
Temporal oscillation frequency chosen by (2.18) to balance the transport error and Newton error; a free construction parameter.
Gamma = ceil((1+delta/3)/(1+2delta/3-beta))
Number of Newton steps needed to reduce the gluing error to the desired size; set in Section 5.2.

assumptions (5)

domain assumption The multipliers m are odd, homogeneous of order delta in [-1,0], smooth away from the origin, and satisfy m(xi).xi = 0.
Defines the class of active scalar equations studied in Section 1.1.
domain assumption The rigid part of Conjecture 1.1 for general odd ASE holds: solutions with Lambda^{-1}theta of class above 1+2delta/3 conserve the Hamiltonian.
This external result of Isett and Ma [23] sets the sharp threshold that the flexible construction must match; cited in the abstract, Remark 1.2, and Section 2.3.
domain assumption The Newton-Nash scheme of Giri and Radu [17] (flow estimates, temporal partitions, stability estimates) can be adapted to the momentum formulation (2.3) of the ASE.
Several lemmas in Sections 5 and 6 are quoted from [17] with proofs said to be 'similar to' that paper; the adaptation is asserted rather than fully re-proved.
standard math Standard Littlewood-Paley theory, Calderon-Zygmund commutator estimates, and the bilinear microlocal Lemma C.1 are valid in the stated form.
Background results used throughout; most are quoted from [2], [4], [10], [17], [19], and [31].
ad hoc to paper The trilinear Fourier multiplier estimate Lemma 4.5 is correct as stated for all -1 <= delta <= 0.
This is a new lemma of this paper whose proof contains numerous omitted 'tedious calculations'; the paper's closure argument depends on it. It is neither a standard result nor fully verified in the text.

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Pith. "Pith review of An Onsager-type Theorem for General 2D Active Scalar Equations." pith.science (2026). https://pith.science/paper/MV4IINOS

@misc{pith2026241211094,
  author       = {Pith},
  title        = {Pith review of: An Onsager-type Theorem for General 2D Active Scalar Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MV4IINOS}},
  note         = {Machine review of arXiv:2412.11094}
}

abstract

This paper concerns the Onsager-type problem for general 2-dimensional active scalar equations of the form: $\partial_t \theta+u\cdot\nabla \theta= 0$, with $u=T[\theta]$ being a divergence-free velocity field and $T$ being a Fourier multiplier operator with symbol $m$. It is shown that if $m$ is a odd and homogeneous symbol of order $\delta$: $m(\lambda\xi)=\lambda^{\delta} m(\xi)$, where $\lambda>0, -1\le\delta\le0$, then there exists a nontrivial temporally compact-supported weak solution $\theta\in C_t^0 C_x^{\frac{2\delta}{3}-}$, which fails to conserve Hamiltonian. This result is sharp since all weak solutions of class $C_t^0C_x^{\frac{2\delta}{3}+}$ will necessarily conserve the Hamiltonian (which is proved by P. Isett and A. Ma in arXiv:2403.08279, 2024.) and thus resolves the flexible part of the generalized Onsager conjecture for general 2D odd active scalar equations. Also, in the appendix, analogous results have been obtained for general 2D and 3D even active scalar equations. The proof is achieved by using convex integration scheme at the level $v=-\nabla^{\perp}\cdot\theta$ together with a Newton scheme recently introduced by V. Giri and R. O. Radu (2D Onsager conjecture: a Newton-Nash iteration. Invent. math. (2024).). Moreover, a novel algebraic lemma and sharp estimates for some complicated trilinear Fourier multipliers are established to overcome the difficulties caused by the generality of the equations.

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

Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation
math.AP 2025-02 conditional novelty 8.0 of 10

Non-uniqueness with forcing for alpha-SQG is established across the full supercritical Sobolev range s < alpha + 2/p, using new smooth compactly supported unstable vortices.

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Works this paper leans on

31 extracted references · 22 canonical work pages · cited by 1 Pith paper

[1]

J. Bear. Dynamics of ﬂuids in porous media. . Courier Dover Publications, 1972

work page 1972
[2]

Buckmaster, C

T. Buckmaster, C. De Lellis, L. Sz´ ekelyhidi, and V. Vicol. Onsager’s conjecture for admissible weak solutions. Comm. Pure Appl. Math., https://doi.org/10.1002/cpa.21781. 201 8

doi:10.1002/cpa.21781
[3]

Buckmaster, N

T. Buckmaster, N. Masmoudi, M. Novack, V. Vicol. Intermittent Convex Integration for the 3D Euler equations. Ann. of Math. Studies , Vol. 217, 2023. Preprint arxiv: 2101.09278

arXiv 2023
[4]

Buckmaster, S

T. Buckmaster, S. Shkoller, and V. Vicol. Nonuniqueness of weak solutions to the SQG equation . Communications on Pure and Applied Mathematics, Vol. LXXII, 1809– 1874, 2019

work page 2019
[5]

and Li, D

Cheng, X., Kwon, H. and Li, D. The Non-uniqueness of Steady-State Weak Solutions to the Surface Quasi-Geostrophic Equations . Commun. Math. Phys. 388, 1281–1295 (2021). https://doi.org/10.1007/s00220-021-04247-z. 68

work page doi:10.1007/s00220-021-04247-z 2021
[6]

Conti, C

S. Conti, C. De Lellis, L. Sz´ ekelyhidi Jr. h-principle and rigidity for C1,α isometric embeddings. Nonlinear partial diﬀerential equations, 83–116, Springer, 2012

work page 2012
[7]

Constantin, W

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

work page 1994
[8]

Constantin, A.J

P. Constantin, A.J. Majda, and E. Tabak. Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity, 7(6):1495–1533, 1994

work page 1994

Show all 31 references

[9]

C /acute.ts1ordoba and F

D. C /acute.ts1ordoba and F. Gancedo. Contour dynamics of incompressible 3-D ﬂuids in a porous med ium with diﬀerent densities. Comm. Math. Phys., 273(2):445–471, 2007

2007
[10]

Giri and R

Mimi Dai and V. Giri and R. O. Radu. An Onsager-type theorem for SQG . arXiv:2407.02582, 2024

2024 arXiv
[11]

De Lellis, and L

C. De Lellis, and L. Sz´ ekelyhidi. The Euler equations as a diﬀerential inclusion . Ann. of Math., Vol.170 No.3: 1417–1436, 2009

2009
[12]

De Lellis, and L

C. De Lellis, and L. Sz´ ekelyhidi. Dissipative continuous Euler ﬂows . Invent. Math., Vol.193 No. 2: 377–407, 2013

2013
[13]

Friedlander, F

S. Friedlander, F. Gancedo, W. Sun, and V. Vicol. On a singular incompressible porous media equation.. Journal of Mathematical Physics, 53(11):1–20, 2012

2012
[14]

arXiv:2407.15884, 2024

Vikram Giri and Ujjwal Koley Non-uniqueness of H¨ older continuous solutions for Inhomo geneous Incompressible Euler ﬂows . arXiv:2407.15884, 2024

2024
[15]

A wavelet-inspiredL3-based convex integration framework for the Euler equations

Vikram Giri and Hyunju Kwon and Matthew Novack. A wavelet-inspiredL3-based convex integration framework for the Euler equations . arXiv:2305.18142, 2023

2023 arXiv
[16]

The L3-based strong Onsager theorem

Vikram Giri and Hyunju Kwon and Matthew Novack. The L3-based strong Onsager theorem . arXiv:2305.18509, 2023

2023 arXiv
[17]

Giri and R

V. Giri and R. O. Radu. The 2D Onsager conjecture: a Newton-Nash iteration . Invent. math. (2024). https://doi.org/10.1007/s00222-024-01291-z

2024 doi
[18]

Held, R.T

I.M. Held, R.T. Pierrehumbert, S.T. Garner, and K.L. Swanson. Surface quasi-geostrophic dynamics.. J. Fluid Mech., 282:1–20, 1995

1995
[19]

Global solutions for the gravity water waves system in 2d

Ionescu, A.D., Pusateri, F. Global solutions for the gravity water waves system in 2d. Invent. math. 199, 653–804 (2015)

2015
[20]

P. Isett. A Proof of Onsager’s Conjecture . Ann. of Math., Vol.188 No.3: 1–93, 2018

2018
[21]

Isett and S.-Z

P. Isett and S.-Z. Looi. A proof of Onsager’s conjecture for the SQG equation . arXiv: 2407.02578, 2024

2024 arXiv
[22]

Isett and A

P. Isett and A. Ma. A direct approach to nonuniqueness and failure of compactne ss for the SQG equation. Nonlinearity, 34(5): 3122–3162, 2021

2021
[23]

Isett and A

P. Isett and A. Ma. On the conservation laws and the structure of the nonlineari ty for SQG and its generalizations. arXiv:2403.08279, 2024

2024 arXiv
[24]

Isett and V

P. Isett and V. Vicol. H¨ older continuous solutions of active scalar equations. Ann. PDE., 1(1): 1–77, 2015

2015
[25]

H.K. Moﬀatt. Magnetostrophic turbulence and the geodynamo. In IUTAM Symposium on Compu- tational Physics and New Perspectives in Turbulence, volume 4 of IU TAM Bookser., pages 339–346. Springer, Dordrecht, 2008

2008
[26]

A. G. Ma. Some Onsager’s conjecture type results for a family of odd ac tive scalar equations . UT Austin. 2022. 69

2022
[27]

Moﬀatt and D.E

H.K. Moﬀatt and D.E. Loper. The magnetostrophic rise of a buoyant parcel in the earth’s c ore. Geophysical Journal International, 117(2):394–402, 1994

1994
[28]

J. Nash. C1 isometric imbeddings. Ann. Math., 60:383–396, 1954

1954
[29]

Novack, and V

M. Novack, and V. Vicol. An Intermittent Onsager Theorem. Invent. Math. , in press, 2023

2023
[30]

L. Onsager. Statistical hydrodynamics. Il Nuovo Cimento (1943-1954) , 6:279–287, 1949

1943
[31]

Com- munications on Pure and Applied Mathematics, 71: 90-162, 2018

Wang, Xuecheng Global Inﬁnite Energy Solutions for the 2D Gravity Water Wav es System. Com- munications on Pure and Applied Mathematics, 71: 90-162, 2018. 70

2018

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

J. Bear. Dynamics of ﬂuids in porous media. . Courier Dover Publications, 1972

work page 1972

[2] [2]

Buckmaster, C

T. Buckmaster, C. De Lellis, L. Sz´ ekelyhidi, and V. Vicol. Onsager’s conjecture for admissible weak solutions. Comm. Pure Appl. Math., https://doi.org/10.1002/cpa.21781. 201 8

doi:10.1002/cpa.21781

[3] [3]

Buckmaster, N

T. Buckmaster, N. Masmoudi, M. Novack, V. Vicol. Intermittent Convex Integration for the 3D Euler equations. Ann. of Math. Studies , Vol. 217, 2023. Preprint arxiv: 2101.09278

arXiv 2023

[4] [4]

Buckmaster, S

T. Buckmaster, S. Shkoller, and V. Vicol. Nonuniqueness of weak solutions to the SQG equation . Communications on Pure and Applied Mathematics, Vol. LXXII, 1809– 1874, 2019

work page 2019

[5] [5]

and Li, D

Cheng, X., Kwon, H. and Li, D. The Non-uniqueness of Steady-State Weak Solutions to the Surface Quasi-Geostrophic Equations . Commun. Math. Phys. 388, 1281–1295 (2021). https://doi.org/10.1007/s00220-021-04247-z. 68

work page doi:10.1007/s00220-021-04247-z 2021

[6] [6]

Conti, C

S. Conti, C. De Lellis, L. Sz´ ekelyhidi Jr. h-principle and rigidity for C1,α isometric embeddings. Nonlinear partial diﬀerential equations, 83–116, Springer, 2012

work page 2012

[7] [7]

Constantin, W

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

work page 1994

[8] [8]

Constantin, A.J

P. Constantin, A.J. Majda, and E. Tabak. Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity, 7(6):1495–1533, 1994

work page 1994