{"id":"b6dcd914-a84c-4f1c-8e5d-795c6ebcf5a0","arxiv_id":"2512.08816","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Critical SQG has H1 norm inflation from large smooth data and small-data norm inflation in supercritical W^{β,p} spaces.","lead":"The critical SQG equation, though globally well-posed in H1, is shown to admit smooth initial data whose H1 norm grows by an arbitrarily large factor over short times. The paper also constructs small-data norm inflation in supercritical W^{β,p} spaces, revealing a form of ill-posedness inside a globally regular PDE.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Periodic small-data norm inflation for β>1 is unproven: (5.4) uses an invalid Lp–W^{1,p} interpolation for fractional W^{β,p}.","rationale":"The reader's weakest assumption identifies exactly the most load-bearing defect in the periodic extension. The central R2 construction appears coherent: the ansatz, stationary-phase estimates, and bootstrap are consistent, and the β=1 periodic case is not affected by the invalid interpolation because the exponent 1−β becomes 0. The concern is load-bearing because Theorem 1.3 on T2 promises small-data norm inflation for exactly the range 1<β<2/p where (5.4) is used, and the manuscript supplies no alternative proof for that range. This warrants the CONDITIONAL verdict: the main ideas may be correct, but the periodic supercritical theorem is incomplete as written. I did not find a more fundamental flaw in the R2 argument; the ε-dependent constants in Proposition 3.2 are consistent with the paper's convention of allowing constants to depend on ε, so I do not press that as a separate attack.","tokens_in":18420,"tokens_out":28302,"duration_ms":264318,"concrete_test":"Independently re-derive (5.4) for 1<β<2/p. Concretely, verify the norm equivalence ∥∑_{n∈Z2} θc(·+2πn)∥_{W^{β,p}(T2)} ≲ ∥θc∥_{W^{β,p}(R2)} for the disjoint-support functions (3.5)-(3.6), or else establish (5.4) by interpolating between Lp and W^{2,p}: ∥θ(0)∥_{W^{β,p}(T2)} ≤ C∥θ(0)∥_{Lp}^{1−β/2} ∥θ(0)∥_{W^{2,p}}^{β/2}, using explicit derivative counts from (3.5)-(3.6). If either route closes for all β∈(1,2/p), the periodic theorem is repairable; if neither does, the T2 result for β>1 should be withdrawn.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 5.2's proof of the periodic small-data bound (5.4) states: 'Using interpolation, we obtain ∥θ(0)∥_{W^{β,p}(T2)} ≲ ∥θ(0)∥_{Lp}^{1−β} ∥θ(0)∥_{W^{1,p}}^β.' This is a valid interpolation inequality only for 0≤β≤1. Theorem 1.3 allows 1<β<2/p, for which 1−β<0; the inequality as written is not a valid interpolation (the convexity exponent would need to be β between 0 and 1). The periodic version of Theorem 1.3 for the genuinely supercritical range β>1 therefore rests on an unproven estimate. The R2 proof does not have this defect: Proposition 3.3's lower bound uses the valid interpolation ∥∇θ2∥_{Lp} ≲ ∥θ2∥_{Lp}^{1−1/β} ∥θ2∥_{W^{β,p}}^{1/β} with exponent 1/β∈(0,1). On T2 one can likely repair (5.4) by proving the norm equivalence ∥periodization∥_{W^{β,p}(T2)} ≍ ∥θc∥_{W^{β,p}(R2)} for compactly supported θc, or by interpolating through W^{2,p}; but no such argument appears. Since Section 5 claims the periodic theorem follows from the R2 construction, this is an omitted proof at a load-bearing point, not merely a cosmetic typo.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18809,"tokens_out":19555,"duration_ms":153399,"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":[{"comment":"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.","section":"§5, Proposition 5.2, Eq. (5.4)"}],"minor_comments":[{"comment":"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.","section":"§1, Theorems 1.1 and 1.3"},{"comment":"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.","section":"§3.2.1"},{"comment":"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.","section":"§4.1, Eq. (4.11) and Lemma 2.1"},{"comment":"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.","section":"§5, after Proposition 5.4"},{"comment":"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.","section":"Appendix A.1, Eq. (A.1)–(A.2)"},{"comment":"There is a typo: 'where u[\\bar{\\theta}_2]) is the approximate velocity' has an extra parenthesis.","section":"§3.4, estimate for F_1"}],"recommendation":"major_revision","confidential_remarks":"The only substantive obstacle is the missing estimate for the periodic small-data bound (5.4) when \\beta>1. The rest of the proof appears sound and the R^2 part is convincing. If the authors supply a valid interpolation argument or a periodization norm equivalence for fractional Sobolev spaces, I would be willing to accept the paper. There are no fitted parameters and no circularity concerns; the main theorems are new and within the scope of the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: the paper has the right target and the main R2 proof looks like a real construction, not a sketch. The new thing is treating critical α=1 dissipation as an error and still getting W^{β,p} small-data growth, plus H1 large-data norm inflation. The ansatz comes from Córdoba–Martínez-Zoroa, but the critical case did not follow from their papers, and the stationary-phase lemma generalized to all s>0 is the workhorse. I read the bootstrap in §4; the estimates are explicit and the bookkeeping is coherent. No fitted parameters, no circular reasoning. Credit where it's due.\n\nSoft spots. The periodic extension is the real problem. Proposition 5.2 derives (5.4) by taking ∥θ(0)∥_{W^{β,p}(T2)} ≲ ∥θ(0)∥_{Lp}^{1−β} ∥θ(0)∥_{W^{1,p}}^β. That is valid interpolation only for 0≤β≤1; the theorem advertises 1<β<2/p, and with β>1 the right side has a negative exponent on Lp. The same 'interpolation' appears in (5.5) with exponent 1/β, which is valid. So the flaw is localized to the initial-data estimate on T2 for β>1. I would not call it cosmetic, because Theorem 1.3 on T2 is asserted for that range. It is probably repairable by proving a norm equivalence for the periodization in fractional Sobolev spaces, or by interpolating through W^{2,p}; the ingredients are in the paper. But as written, the periodic half of Theorem 1.3 is unproven.\n\nOther concerns are minor. §3.2.1's heuristic says both leading quadratic errors vanish; the rigorous estimates show only the leading contribution cancels, which is fine, but the prose shouldn't be taken literally. There are a few notational slips, none load-bearing. I didn't find a problem with the citations; the method is attributed, and the authors' own prior references aren't doing any heavy lifting.\n\nWho this is for: anyone working on SQG norm inflation/ill-posedness or critical dissipative active scalars. If the periodic gap is patched, it deserves publication; even now, the R2 theorem is the main prize and looks solid. I'd send it to a serious referee and expect them to ask for the patch before acceptance.","headline":"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.","tokens_in":19287,"tokens_out":4675,"would_cite":true,"duration_ms":48117,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","35B30","35R11"],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["norm inflation","critical SQG","data-to-solution map","uniform boundedness","fractional Sobolev spaces","supercritical spaces","ill-posedness","critical dissipation"],"falsifier":"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.","tokens_in":18331,"feed_emoji":"📈","tokens_out":8738,"duration_ms":71990,"temperature":0.7,"pith_summary":"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<p<2 and 1≤β<2/p, where arbitrarily small initial data grow to order ε^{-1}. The proof works by building a two-scale approximate solution—a concentrated radial vortex plus a rapidly oscillating angular wave carried by its velocity—and showing that the true solution tracks this approximation long enough for the growth to occur. If correct, this shows that global well-posedness in H1 does not imply any uniform control of the solution map in that same topology, a phenomenon invisible to the L∞ maximum principle.","feed_headline":"Critical SQG solutions inflate H1 norm despite global regularity","feed_subtitle":"Tiny data in slightly subcritical spaces grow by 1/ε in time ≤ε, so H1 solution map is unbounded.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["SQG norm inflation: small data, huge H1 growth","Critical SQG: H1 solution map not uniformly bounded","Tiny SQG data can inflate H1 norm by 1/ε","Norm inflation in critical SQG despite global regularity"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["SQG norm inflation: small data, huge H1 growth","Critical SQG: H1 solution map not uniformly bounded","Tiny SQG data can inflate H1 norm by 1/ε","Norm inflation in critical SQG despite global regularity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000927,"raw_usage":{"total_tokens":3772,"prompt_tokens":672,"completion_tokens":3100,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":416,"completion_tokens_details":{"reasoning_tokens":3041}},"tokens_in":416,"tokens_out":3100,"duration_ms":20706,"temperature":1.0,"reasoning_tokens":3041,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T17:36:02.008520+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}