Pith. sign in

REVIEW 1 major objections 6 minor 38 references

Norm Inflation For The Critical SQG Equation

T0 review · 1 major / 6 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read Even though the critical surface quasi-geostrophic equation is globally well-posed in the Sobolev space H1, this paper proves that the data-to-solution map is not uniformly bounded: solutions can inflate their H1 norm by an arbitrarily larg

desk verdict The R2 construction is serious and likely right, but the periodic small-data theorem for β>1 rests on invalid interpolation and needs a patch. read the letter →

arxiv 2512.08816 v2 pith:6IJYROVL submitted 2025-12-09 math.AP

Dengjun Guo , Xiaoyutao Luo This is my paper · ORCID
classification math.AP MSC 35Q3535B3035R11
keywords norminflationcriticalSQGdata-to-solutionmapuniformboundednessfractionalSobolevspacessupercriticalill-posednessdissipation
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

Even though the critical surface quasi-geostrophic (SQG) equation is globally well-posed in the Sobolev space H1, this paper proves that its data-to-solution map is not uniformly bounded: for any ε>0 there are smooth, compactly supported initial data whose H^{1-ε} norm is at most ε, and a time T≤ε, such that the H1 norm grows by at least a factor of 1/ε by time T. The construction also yields small-data norm inflation in the supercritical fractional Sobolev spaces W^{β,p} for 1

What carries the argument

The central object is the two-scale approximate solution θ=θ1+θ2, where θ1=ε λ^{2/p-β} f(λr) is a stationary radial bump and θ2=ε λ^{2/p-β} N^{-β} g(λr) cos(Nα - ε N t λ^{1+2/p-β} h(λr)) is a rapidly oscillating angular wave transported by the velocity of θ1. The key identity is Theorem 2.2: for such oscillatory functions, Λ^{-s}θ ≈ c_s N^{-s} θ (r^{-2}+|h'(r)|^2)^{s/2} with error decaying like N^{-1-s}. This converts the nonlocal fractional operator into an explicit local leading term, letting the authors show that the dominant errors cancel and that the dissipation is a small error at critical regularity. The remainder η is then controlled by an energy/bootstrap estimate up to the critical

What would settle it

Resolve the critical SQG equation on T^2 with the paper's constructed initial data at high spatial resolution and measure ∥θ(T)∥_{H^1}/∥θ(0)∥_{H^1} at the predicted time T=λ^{β-1-2/p} ε^{-1-2/β}; if the ratio stays bounded as ε→0, the ansatz fails. More directly, compute the fractional Sobolev norm ∥θ(0)∥_{W^{β,p}(T^2)} of the periodized initial data for 1<β<2/p; the proof's estimate (5.4) relies on an interpolation inequality that is only valid for β≤1, so a direct norm computation would settle whether the periodic small-data theorem is true as stated.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.1 (and its companion Theorem 1.3): on R^2 or T^2, for every ε>0 there exists a smooth, compactly supported solution of the critical SQG equation and a time 0<T≤ε such that ∥θ0∥_{H^{1-ε}}≤ε and ∥θ(T)∥_{H^1}/∥θ0∥_{H^1} ≥ ε^{-1}. In other words, the H1 data-to-solution map exists globally but is not uniformly bounded. The engine of the proof is an explicit approximate solution in which a stationary radial profile θ1 generates a large sweeping velocity, transporting a high-frequency angular profile θ2; the fractional dissipation and the nonlinear interactions are shown to be subdominant, thanks to a local approximation for Λ^{-s} acting on oscillatory funct

Load-bearing premise

The periodic small-data theorem for 1<β<2/p rests on the interpolation inequality (5.4) in Proposition 5.2, which is valid only for 0≤β≤1, and the paper provides no direct estimate for the fractional Sobolev norm of the periodized construction when β>1.

Editorial extensions

If this is right

  • The H1 data-to-solution map for critical SQG is not uniformly bounded, even though the equation is globally well-posed in H1.
  • Arbitrarily small initial data in W^{β,p} (1<p<2, 1≤β<2/p) can produce solutions whose W^{β,p} norm reaches at least ε^{-1} at some time T≤ε.
  • Both behaviors hold on the plane and on the torus, via periodic extension of the same construction.
  • Small-data norm inflation in supercritical Sobolev spaces H^s with s<1 remains open; the present method requires β≥1 to control the remainder.

Reading between the lines

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

  • If the construction is correct, numerical schemes for critical SQG that assume uniform H1 stability will under-predict growth near the constructed data; running a spectral simulation with the paper's ansatz and measuring the H1 growth ratio at the predicted time would be a direct check.
  • The transport-amplification mechanism is likely portable to other critical dissipative active-scalar equations with the same scaling symmetry, wherever an analogue of the local Λ^{-s} approximation holds.
  • The flagged interpolation gap in the periodic proof for β>1 (inequality (5.4)) is a concrete place to look for a repair: a direct frequency-localized estimate for the periodized profile's fractional Sobolev norm would either close the theorem or reveal a genuine restriction.
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.

Referee Report

1 major / 6 minor

Summary. The paper studies the critical dissipative surface quasi-geostrophic equation (1.1) on R^2 and T^2, and proves norm-inflation results despite global H^1 well-posedness. The construction is an explicit two-scale approximate solution \bar{\theta}=\bar{\theta}_1+\bar{\theta}_2, where \bar{\theta}_1 is a stationary radial profile and \bar{\theta}_2 is a rapidly oscillating profile transported by the radial flow. The dissipation is treated as an error term using a stationary-phase lemma for \Lambda^{-s} on oscillatory functions (Theorem 2.2). A bootstrap argument (Theorem 4.1) shows that the exact solution with the same initial data remains close to the approximate solution up to the inflation time. The authors conclude large-data H^1 norm inflation (Theorem 1.1) and small-data W^{\beta,p} norm inflation for 1<p<2, 1\le\beta<2/p (Theorem 1.3), on both the whole space and the torus.

Significance. If correct, the results are significant: they show that global well-posedness of critical SQG in H^1 does not imply uniform boundedness of the data-to-solution map, and they establish genuine small-data ill-posedness in supercritical W^{\beta,p} spaces. The construction is explicit and parameter-free, with no fitted constants, and the main technical achievement—controlling the critical dissipation via a refined oscillatory-integral estimate—is novel and well executed. The proof on R^2 is detailed and internally consistent. The periodic extension, however, contains a load-bearing gap for the range \beta>1 that, as written, leaves part of Theorem 1.3 unproved.

major comments (1)
  1. [§5, Proposition 5.2, Eq. (5.4)] The proof of (5.4) states: 'Using interpolation, we obtain \|\theta(0)\|_{W^{\beta,p}(\mathbb{T}^2)} \lesssim \|\theta(0)\|_{L^p}^{1-\beta}\|\theta(0)\|_{W^{1,p}}^\beta.' This interpolation inequality is valid only for 0\le\beta\le1. Since Theorem 1.3 allows 1<\beta<2/p, the periodic small-data theorem is unproven for that range. The lower bound (5.5) uses the valid interpolation with exponent 1/\beta, but the upper bound (5.4) does not. A repair is likely available, for instance by interpolating between L^p and W^{2,p} (using the fact that \beta<2) or by proving a periodization norm equivalence for compactly supported functions, but no such argument appears. Because Section 5 claims the periodic theorem follows from the R^2 construction, this is a missing proof at a load-bearing point, not a cosmetic typo.
minor comments (6)
  1. [§1, Theorems 1.1 and 1.3] The statement '\theta_0\in C_c^\infty(\mathbb{R}^2)' appears even when \Omega=\mathbb{T}^2. In the periodic case the initial data should be a smooth periodic function, e.g. the periodization of the compactly supported R^2 profile; please clarify.
  2. [§3.2.1] The heuristic sentence 'both the leading terms of u[\bar{\theta}_2]\cdot\nabla\bar{\theta}_1 and u[\bar{\theta}_2]\cdot\nabla\bar{\theta}_2 vanish' is inaccurate: the estimates in §3.4 show that these terms are small after replacing u[\bar{\theta}_2] by the approximate velocity \bar{u}[\bar{\theta}_2], not that the original terms vanish. Please rephrase to avoid confusion.
  3. [§4.1, Eq. (4.11) and Lemma 2.1] The parenthetical 'setting -\delta_1=\delta_2=2/p=1/2' is confusing: the p in 2/p here is not the original integrability exponent but the auxiliary p=4 used in the W^{2,4}–W^{1,4} interpolation. Please write the interpolation parameters explicitly.
  4. [§5, after Proposition 5.4] The statement 'Then argued as in Theorem 4.1, we can obtain the estimate for the remainder term' is too terse. The periodic bootstrap involves additional nonlocal contributions from periodic images, and although Lemma 5.3 provides the needed control, the analogue of Theorem 4.1 should be stated and proved, or at least the differences should be spelled out.
  5. [Appendix A.1, Eq. (A.1)–(A.2)] The chain of embeddings in the proof of Lemma 2.1 uses the symbol s both in the Besov and Sobolev embeddings; the roles of s_1 and s_2 should be stated explicitly to avoid ambiguity.
  6. [§3.4, estimate for F_1] There is a typo: 'where u[\bar{\theta}_2]) is the approximate velocity' has an extra parenthesis.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: explicit approximate-solution construction with independent error and stability estimates.

full rationale

I walked the derivation chain: the ansatz (1.6), the definition of the approximate solution θ = θ1 + θ2, the error decomposition F = F1+F2+F3+F4, the parameter choices (3.1)-(3.2), the lower bounds in Proposition 3.3, and the bootstrap/stability argument in Theorem 4.1. No step reduces to its own input. The norm-inflation lower bounds are computed directly from the explicit profile of θ2(T); the remainder η is bounded by independent Gronwall/bootstrap estimates using the Córdoba–Córdoba maximum principle and interpolation. There are no fitted parameters, no data-to-parameter inversion, and no uniqueness theorem imported from the authors' prior work. The citations to [13,14] describe the general multi-scale framework and are not used as a substitute for the proofs here; the authors' own prior papers [31,32] are cited only as related background and are not load-bearing. One non-circular correctness concern should be flagged: in Proposition 5.2, the proof of (5.4) states 'Using interpolation, we obtain ∥θ(0)∥_{W^{β,p}(T2)} ≲ ∥θ(0)∥_{Lp}^{1−β}∥θ(0)∥_{W^{1,p}}^β,' but the interpolation inequality in that form is valid only for 0≤β≤1, whereas Theorem 1.3 allows 1<β<2/p. This is an omitted estimate/gap in the periodic extension for β>1, not a circular reduction: the theorem is not assumed in its own proof and the R2 argument does not rely on the invalid step. Because the central derivation is self-contained and externally checkable, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No parameters are fitted to data; λ,N,T are set by the explicit formulas (3.1)–(3.2), and the profiles f,g,h are chosen qualitatively (Lemma A.1). No new physical entities are introduced; θbar, θ1, θ2 are explicit trial functions, and f,g,h are constructed profiles.

assumptions (3)
  • domain assumption Critical SQG is globally well-posed in H1 (Dong [22]; Dong–Du [23]).
    Used in Section 4 to define the exact solution θ with the constructed smooth initial data and to justify the difference equation (4.1).
  • standard math Córdoba–Córdoba inequality: for the fractional Laplacian, ∫ |η|^{q-2} η Λη ≥ 0 (Lemma 2.5 of [11]).
    Used in Theorem 4.1 Step 1 to discard the dissipative term in the L^q estimate of η.
  • standard math Besov embedding and interpolation from Bahouri–Chemin–Danchin [1].
    Used in Lemma 2.1 and in the bootstrap closing estimate (4.11)–(4.12).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Norm Inflation For The Critical SQG Equation." pith.science (2026). https://pith.science/paper/6IJYROVL

@misc{pith2026251208816,
  author       = {Pith},
  title        = {Pith review of: Norm Inflation For The Critical SQG Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6IJYROVL}},
  note         = {Machine review of arXiv:2512.08816}
}
abstract

We consider the critical dissipative surface quasi-geostrophic (SQG) equation on $\mathbb{R}^2$ or $\mathbb{T}^2$. Despite global regularity of the equation, we show that the data-to-solution map at the critical level $H^1$ is not uniformly bounded. We construct solutions that experience $H^1$ norm inflation from smooth, compactly supported initial data with large $H^1$ norm. We also demonstrate small-data norm inflation in supercritical Sobolev spaces $W^{\beta,p}$ for $1<p<2$ and $1\le\beta<\tfrac{2}{p}$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

38 extracted references · 3 linked inside Pith

  1. [1]

    343, Springer, Heidelberg, 2011

    Hajer Bahouri, Jean-Yves Chemin, and Rapha¨ el Danchin,Fourier analysis and nonlinear partial differential equations, Grundlehren der mathematis- chen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343, Springer, Heidelberg, 2011. MR 2768550

    2011

  2. [2]

    Math.201(2015), no

    Jean Bourgain and Dong Li,Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces, Invent. Math.201(2015), no. 1, 97–

    2015

  3. [3]

    Pure Appl

    Tristan Buckmaster, Steve Shkoller, and Vlad Vicol,Nonuniqueness of weak solutions to the SQG equation, Comm. Pure Appl. Math.72(2019), no. 9, 1809–1874. MR 3987721

    2019

  4. [4]

    Caffarelli and Alexis Vasseur,Drift diffusion equations with frac- tional diffusion and the quasi-geostrophic equation, Ann

    Luis A. Caffarelli and Alexis Vasseur,Drift diffusion equations with frac- tional diffusion and the quasi-geostrophic equation, Ann. of Math. (2)171 (2010), no. 3, 1903–1930. MR 2680400

    2010

  5. [5]

    Carrillo and Lucas C

    Jos´ e A. Carrillo and Lucas C. F. Ferreira,The asymptotic behaviour of subcritical dissipative quasi-geostrophic equations, Nonlinearity21(2008), no. 5, 1001–1018. MR 2412324

    2008

  6. [6]

    Dif- ferential Equations443(2025), Paper No

    Young-Pil Choi, Jinwook Jung, and Junha Kim,On well/ill-posedness for the generalized surface quasi-geostrophic equation in H¨ older spaces, J. Dif- ferential Equations443(2025), Paper No. 113521, 36. MR 4919126

    2025

  7. [7]

    PDE7 (2021), no

    Maria Colombo and Silja Haffter,Estimate on the dimension of the singu- lar set of the supercritical surface quasigeostrophic equation, Ann. PDE7 (2021), no. 1, Paper No. 6, 62. MR 4235801

    2021

  8. [8]

    Peter Constantin, Diego Cordoba, and Jiahong Wu,On the critical dissipa- tive quasi-geostrophic equation, Indiana Univ. Math. J.50(2001), 97–107, Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000). MR 1855665

    2001

Show all 38 references
  1. [9]

    Majda, and Esteban Tabak,Formation of strong fronts in the2-D quasigeostrophic thermal active scalar, Nonlinearity 7(1994), no

    Peter Constantin, Andrew J. Majda, and Esteban Tabak,Formation of strong fronts in the2-D quasigeostrophic thermal active scalar, Nonlinearity 7(1994), no. 6, 1495–1533. MR 1304437 26

    1994

  2. [10]

    Peter Constantin and Jiahong Wu,Behavior of solutions of 2D quasi- geostrophic equations, SIAM J. Math. Anal.30(1999), no. 5, 937–948. MR 1709781

    1999

  3. [11]

    Antonio C´ ordoba and Diego C´ ordoba,A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys.249(2004), no. 3, 511–

    2004

  4. [12]

    8, Paper No

    Diego C´ ordoba, Jos´ e Lucas-Manch´ on, and Luis Mart ´ ınez-Zoroa,Strong ill- posedness and non-existence in Sobolev spaces for generalized-SQG, Non- linearity38(2025), no. 8, Paper No. 085021, 57. MR 4950574

    2025

  5. [13]

    Math.407(2022), Paper No

    Diego C´ ordoba and Luis Mart ´ ınez-Zoroa,Non existence and strong ill- posedness inC k and Sobolev spaces for SQG, Adv. Math.407(2022), Paper No. 108570, 74. MR 4452676

    2022

  6. [14]

    PDE10(2024), no

    ,Global unique solutions with instantaneous loss of regularity for SQG with fractional diffusion, Ann. PDE10(2024), no. 2, Paper No. 21,

    2024

  7. [15]

    ,Non-existence and strong ill-posedness inC k,β for the generalized surface quasi-geostrophic equation, Comm. Math. Phys.405(2024), no. 7, Paper No. 170, 53. MR 4768540

    2024

  8. [16]

    O˙ za´ nski,Instanta- neous gap loss of Sobolev regularity for the 2D incompressible Euler equa- tions, Duke Math

    Diego C´ ordoba, Luis Mart ´ ınez-Zoroa, and Wojciech S. O˙ za´ nski,Instanta- neous gap loss of Sobolev regularity for the 2D incompressible Euler equa- tions, Duke Math. J.173(2024), no. 10, 1931–1971. MR 4776418

    2024

  9. [17]

    Math.481(2025), Paper No

    ,Instantaneous continuous loss of regularity for the SQG equation, Adv. Math.481(2025), Paper No. 110553, 49. MR 4963986

    2025

  10. [18]

    Michele Coti Zelati and Vlad Vicol,On the global regularity for the super- critical SQG equation, Indiana Univ. Math. J.65(2016), no. 2, 535–552. MR 3498176

    2016

  11. [19]

    Michael Dabkowski,Eventual regularity of the solutions to the supercritical dissipative quasi-geostrophic equation, Geom. Funct. Anal.21(2011), no. 1, 1–13. MR 2773101

    2011

  12. [20]

    PDE7(2014), no

    Michael Dabkowski, Alexander Kiselev, Luis Silvestre, and Vlad Vicol, Global well-posedness of slightly supercritical active scalar equations, Anal. PDE7(2014), no. 1, 43–72. MR 3219499

    2014

  13. [21]

    5, 1525–1535

    Michael Dabkowski, Alexander Kiselev, and Vlad Vicol,Global well- posedness for a slightly supercritical surface quasi-geostrophic equation, Nonlinearity25(2012), no. 5, 1525–1535. MR 2921019

    2012

  14. [22]

    Hongjie Dong,Dissipative quasi-geostrophic equations in critical Sobolev spaces: smoothing effect and global well-posedness, Discrete Contin. Dyn. Syst.26(2010), no. 4, 1197–1211. MR 2600742 27

    2010

  15. [23]

    Hongjie Dong and Dapeng Du,Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space, Discrete Contin. Dyn. Syst.21(2008), no. 4, 1095–1101. MR 2399451

    2008

  16. [24]

    Hongjie Dong and Dong Li,Spatial analyticity of the solutions to the sub- critical dissipative quasi-geostrophic equations, Arch. Ration. Mech. Anal. 189(2008), no. 1, 131–158. MR 2403602

    2008

  17. [25]

    Elgindi and Nader Masmoudi,L ∞ ill-posedness for a class of equations arising in hydrodynamics, Arch

    Tarek M. Elgindi and Nader Masmoudi,L ∞ ill-posedness for a class of equations arising in hydrodynamics, Arch. Ration. Mech. Anal.235(2020), no. 3, 1979–2025. MR 4065655

    2020

  18. [26]

    PDE17(2024), no

    In-Jee Jeong and Junha Kim,Strong ill-posedness for SQG in critical Sobolev spaces, Anal. PDE17(2024), no. 1, 133–170. MR 4702316

    2024

  19. [27]

    O˙ za´ nski,Instanta- neous continuous loss of Sobolev regularity for the 3D incompressible Euler equation, arXiv:2508.06333, 2025

    In-Jee Jeong, Luis Mart ´ ınez-Zoroa, and Wojciech S. O˙ za´ nski,Instanta- neous continuous loss of Sobolev regularity for the 3D incompressible Euler equation, arXiv:2508.06333, 2025

    2025 arXiv

  20. [28]

    Ning Ju,Dissipative 2D quasi-geostrophic equation: local well-posedness, global regularity and similarity solutions, Indiana Univ. Math. J.56(2007), no. 1, 187–206. MR 2305934

    2007

  21. [29]

    Junha Kim and In-Jee Jeong,A simple ill-posedness proof for incompress- ible Euler equations in critical Sobolev spaces, J. Funct. Anal.283(2022), no. 10, Paper No. 109673, 34. MR 4470735

    2022

  22. [30]

    Kiselev, F

    A. Kiselev, F. Nazarov, and A. Volberg,Global well-posedness for the crit- ical 2D dissipative quasi-geostrophic equation, Invent. Math.167(2007), no. 3, 445–453. MR 2276260

    2007

  23. [31]

    Xiaoyutao Luo,Illposedness of incompressible fluids in supercritical sobolev spaces, arXiv:2404.07813, 2024

    2024 arXiv

  24. [32]

    ,Sharp norm inflation for 3d navier-stokes equations in supercritical spaces, arXiv:2504.08288, 2025

    2025 arXiv

  25. [33]

    Fabien Marchand,Existence and regularity of weak solutions to the quasi- geostrophic equations in the spacesL p or ˙H −1/2, Comm. Math. Phys.277 (2008), no. 1, 45–67. MR 2357424

    2008

  26. [34]

    Hideyuki Miura,Dissipative quasi-geostrophic equation for large initial data in the critical Sobolev space, Comm. Math. Phys.267(2006), no. 1, 141–

    2006

  27. [35]

    Resnick,Dynamical problems in non-linear advective partial differ- ential equations, ProQuest LLC, Ann Arbor, MI, 1995, Thesis (Ph.D.)–The University of Chicago

    Serge G. Resnick,Dynamical problems in non-linear advective partial differ- ential equations, ProQuest LLC, Ann Arbor, MI, 1995, Thesis (Ph.D.)–The University of Chicago. MR 2716577 28

    1995

  28. [36]

    Luis Silvestre,Eventual regularization for the slightly supercritical quasi- geostrophic equation, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire27 (2010), no. 2, 693–704. MR 2595196

    2010

  29. [37]

    Jiahong Wu,Dissipative quasi-geostrophic equations withL p data, Electron. J. Differential Equations (2001), No. 56, 13. MR 1846672

    2001

  30. [38]

    Differential Equations 253(2012), no

    Liutang Xue and Xiaoxin Zheng,Note on the well-posedness of a slightly supercritical surface quasi-geostrophic equation, J. Differential Equations 253(2012), no. 2, 795–813. MR 2921214 29

    2012

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.