{"id":"cc3ebd07-af54-480c-982e-17d8e1da9780","arxiv_id":"2502.10274","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"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.","lead":"This paper proves that the forced generalized surface quasi-geostrophic equations, including 2D Euler and SQG, admit many weak solutions from zero initial data at all supercritical Sobolev regularities. It also constructs smooth unstable vortices and, by time reversal, global smooth solutions that are neither rotating nor traveling.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The α=1 regularization rests on Proposition 5.2, whose proof is omitted; without it the SQG branch of Theorem 1.1 is unsupported.","rationale":"The reader identified Proposition 5.2 as the weakest assumption, and the manuscript confirms this: the proof is explicitly omitted, the α=1 case is singled out as reversing the roles of the operators, and no independent verification is provided. I agree that this is the single most load-bearing concern, since every subsequent step—self-similar instability, nonlinear instability, and the final forced non-uniqueness for α=1—uses the regularized SQG vortex. The concern is not that the result is false, only that a key existence step is currently asserted rather than demonstrated. The concrete test I propose would distinguish a genuine mathematical gap from a merely expositional omission. I do not see a stronger objection: Section 4's discriminant argument is detailed, the compactness obstruction for α=1 is addressed by the skew-adjoint decomposition in Section 6.6, and the nonlinear energy estimates in Section 7 are presented at length. Thus the reader's CONDITIONAL verdict is appropriate and unchanged.","tokens_in":68072,"tokens_out":2546,"duration_ms":29252,"concrete_test":"Independently supply the omitted contraction proof for Proposition 5.2: write the fixed-point map F_ε defined by (5.16), fix M > ∥(f0, y0, γ0)∥_{L2(I)^2 × C}, and prove for small ε that F_ε maps the ball B_M into itself and is a contraction, using explicit uniform bounds for B0 (Lemma 5.3) and B1 (Lemma 5.6), plus the zero-mean compatibility of A1. If the contraction cannot be closed, the SQG regularization is actually in doubt; if it closes, the gap is only expositional and the verdict should stand.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim includes α=1 (SQG), and every later theorem depends on the smooth unstable vortex from Theorem 3.3. That vortex is produced in Section 5.4, where the SQG regularization is genuinely different from the α<1 case: Lemma 5.6 gives Aε_1 = (log ε)A1 + B1, and the paper decomposes g = μ + f/log ε to derive the coupled system (5.15)–(5.16). Proposition 5.2 then asserts the existence of (fε, yε, γε) for small ε, with the text: 'The proof is analogous, and we therefore omit the details.' This is not a cosmetic omission: the fixed-point map in (5.16) couples f, y, and γ through the compatibility condition for the zero-mean operator A1, and the contraction must control the log ε terms with uniform bounds from Lemmas 5.3 and 5.6. If the contraction fails, or if the compatibility condition cannot be stably inverted, the smooth SQG vortex does not exist and Theorems 3.4, 3.5, and 1.1 for α=1 collapse. Since no alternative proof or numerical verification is supplied, this is the most load-bearing weakness. The additional issue that Corollary 1.3 promises 'global smooth' but establishes only fixed H^m is real but secondary.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims non-uniqueness for the forced α-SQG equation (0 ≤ α ≤ 1) in the full supercritical Sobolev regime s < α + 2/p, with forcing in L^1([0,T], W^{s,p} ∩ Ḣ^{(α−2)/2}). The proof follows Vishik's instability strategy: the authors construct a piecewise constant unstable vortex, regularize it to a compactly supported smooth vortex, prove self-similar linear instability via semigroup perturbation, and then prove nonlinear instability via weighted energy estimates. The same machinery yields smooth unstable vortices and, by time reversal, global H^m solutions to the unforced α-SQG equation that are neither rotating nor traveling. The central theorem (Theorem 1.1) and its refinements (Theorems 3.1, 3.4, 3.5) all rest on Theorem 3.3, whose proof in the case α = 1 depends on Proposition 5.2.","tokens_in":68330,"tokens_out":7232,"duration_ms":73543,"significance":"If the full proof can be verified, this is a substantial result: it would establish the sharp Sobolev threshold for non-uniqueness with forcing for the entire α-SQG family, including the SQG case α = 1, and it would provide the first rigorous construction of unstable vortices for 0 < α ≤ 1. The paper is rich in technical content: the kernel asymptotics for I_{n,α} near σ = 1, the discriminant computation for the piecewise constant vortex, the semigroup perturbation argument, and the inductive weighted energy estimates are all presented in considerable detail. The ad hoc definition of the force in (3.6) is a legitimate degree of freedom for the forced non-uniqueness statement, not circular reasoning. However, the α = 1 regularization step contains a load-bearing omitted proof, and one headline corollary is stated more strongly than what is proved.","major_comments":[{"comment":"Proposition 5.2 is the crucial bridge for the SQG case α = 1, but its proof is omitted with the sentence 'The proof is analogous, and we therefore omit the details.' This is not a cosmetic omission. For α = 1 the expansion in Lemmas 5.3 and 5.6 gives A_0^ε = A_0 + εB_0 and A_1^ε = (log ε)A_1 + B_1, and after the ansatz g = μ + f/log ε the stability equation becomes the coupled system (5.15)–(5.16). The fixed-point map in (5.16) links f, y, and γ through the compatibility condition for the zero-mean operator A_1, and the contraction must control the log ε factors using uniform bounds from Lemmas 5.3 and 5.6. This is exactly where the log divergence of the α = 1 kernel must be tamed, and it is not the same smallness structure as in the α < 1 case of Proposition 5.1, where the error is ε^{1−α}/(1−α). Since Theorems 3.4, 3.5, and ultimately Theorem 1.1 for α = 1 all rely on the smooth vortex from Theorem 3.3, the omitted proof is load-bearing. The authors should supply the full fixed-point argument, either in the main text or in an appendix, including the verification of uniform contraction and the stable inversion of the compatibility condition.","section":"§5.4, Proposition 5.2"},{"comment":"The abstract and Corollary 1.3 describe the by-product as 'global smooth solutions,' but the theorem only establishes θ ∈ C([0,∞), H^m ∩ Ḣ^{(α−2)/2}) for a fixed m > α + 1. The authors themselves note in the paragraph after Corollary 1.3 that extension to the global C^∞ case is open. The wording should be corrected to 'global H^m solutions for each fixed m' in the abstract, the introduction, and Corollary 1.3, so that the statement does not overclaim regularity.","section":"§1.4, Corollary 1.3 and abstract"}],"minor_comments":[{"comment":"In the first line of the proof, the expression ε∂_r θ̄_ε = ε(c_1δ_{r_1} + c_1δ_{r_1}) ∗ η_ε should read ε(c_1δ_{r_1} + c_2δ_{r_2}) ∗ η_ε; the second coefficient is a typo.","section":"§5.1, Lemma 5.2"},{"comment":"In the displayed formula for ∂^J( V̄ · ∇Θ), the angular derivative on Θ is written as ∂^{j_1+1}_φ; it should be ∂^{j_2+1}_φ, consistently with ∂^J = ∂^{j_1}_R ∂^{j_2}_φ.","section":"§7.5, Proposition 7.2 proof"},{"comment":"The sentence 'for every k ∈ N, the other solutions θ_ε can be upgraded to be in C^k_c for positive times' should be qualified as 'for positive times in the interval of construction, with estimates degenerating as t → 0,' since the paper explicitly notes that the estimates deteriorate as t → 0.","section":"§1.1, paragraph on upgrades"},{"comment":"The presentation of the Golovkin trick is clear, but it would help to state explicitly that the two solutions Θ^+ and Θ^− have the same initial data at τ = −∞ and the same forcing G, since that is the point of the construction.","section":"§3.3, Golovkin trick"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read the Castro-Faraco-Mengual-Solera SQG paper. The new result is real: for every 0 <= alpha <= 1 and s < alpha + 2/p, the forced alpha-SQG equation has uncountably many weak solutions in W^{s,p} starting from zero, with the force in the natural energy space. They also construct the first rigorous smooth compactly supported unstable vortices for 0 < alpha <= 1, and obtain global H^m solutions converging to a vortex as t -> infinity. This is a substantial step in the Vishik program.\n\nA lot of the proof is done carefully. The piecewise-constant vortex uses detailed Gamma-function asymptotics, the discriminant calculation is explicit, and the noncompactness of K at alpha = 1 is handled by splitting it into a skew-adjoint part plus a compact commutator. That decomposition looks right and is genuinely new. The weighted energy estimates in Section 7 are technical but coherent, and the paper is honest about what it does and does not prove.\n\nThe soft spot is Proposition 5.2. The alpha = 1 regularization step needs a fixed point for the coupled system (5.15)-(5.16), and the proof is omitted with \"the proof is analogous.\" This is not a cosmetic omission. The operator A_1 has a zero-mean compatibility constraint, y and gamma are determined through that condition, and the log-epsilon terms need uniform control. If that fixed point fails, the smooth vortex used by every later theorem does not exist for SQG. I do not claim it fails--the structure is similar to the alpha < 1 case--but as written the SQG branch of Theorem 1.1 is unsupported. A referee should ask for the details or a correct reference.\n\nMinor point: the abstract and Corollary 1.3 say \"global smooth\" but the theorem gives H^m for fixed m > alpha + 1; the authors themselves note that C^infty is open. That wording should be softened.\n\nBottom line: this deserves a serious referee. The main theorem is important and most of the machinery is in the paper. I would send it to a good journal, request a major revision that fills Proposition 5.2 and fixes the \"smooth\" overstatement, and accept after that.","headline":"Forced alpha-SQG non-uniqueness with Sobolev regularity is a real result, and most of the proof is solid, but the alpha=1 branch leans on an omitted fixed-point proof that a referee must fill.","tokens_in":68884,"tokens_out":3009,"would_cite":true,"duration_ms":30870,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","35Q31","35A02","76B03","35B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"For the forced $\\alpha$-SQG family, Sobolev non-uniqueness holds exactly below the line $s = \\alpha + 2/p$.","keywords":["alpha-SQG equation","non-uniqueness","unstable vortices","self-similar instability","supercritical Sobolev spaces","forced 2D Euler","global smooth solutions","Vishik method"],"falsifier":"Compute the discriminant $\\Delta(\\sigma)$ of Lemma 4.6 for $\\alpha=1$ and $n=2$ by numerical quadrature of the kernels $I_{1,1}$ and $I_{2,1}$ on $\\sigma\\in(1/2,1)$; Proposition 4.1 predicts $\\Delta(1)=\\Delta'(1)=0$ and $\\Delta''(1)<0$, so $\\Delta<0$ for $\\sigma$ just below 1, and finding $\\Delta\\ge0$ on a sequence tending to 1 would refute the claimed unstable vortex, as would an explicit failure of the contraction in Proposition 5.2 for $\\alpha=1$.","tokens_in":67827,"feed_emoji":"🌀","tokens_out":9534,"duration_ms":90907,"temperature":0.7,"pith_summary":"This paper proves that adding a fixed force to the generalized surface quasi-geostrophic equations ($\\alpha$-SQG, the family interpolating between 2D Euler at $\\alpha=0$ and the surface quasi-geostrophic equation at $\\alpha=1$) destroys uniqueness in a sharp way: for every $0\\le\\alpha\\le1$ and every Sobolev exponent pair with $s<\\alpha+2/p$, there is a force for which uncountably many weak solutions start from zero. The mechanism is a new class of smooth, compactly supported unstable vortices whose linearized dynamics have an exponentially growing mode; in self-similar coordinates this mode becomes a genuine second solution, and careful energy estimates show the nonlinear correction cannot kill it. If the result is correct, the forced version of the conjectured uniqueness threshold is settled at natural sharpness for the whole family, and the same construction reverses in time to give global smooth unforced solutions that are neither rotating nor traveling.","feed_headline":"Unstable vortices break uniqueness for forced SQG flows","feed_subtitle":"Below one critical smoothness line, one force admits uncountably many different flows.","key_machinery":"The central object is a self-similarly nonlinearly unstable vortex: a smooth compactly supported radial vortex $\\bar\\Theta$ such that, in the self-similar variables $\\tau=\\frac{1}{ab}\\log t$, $X=x/(abt)^{1/a}$, the linearization $L_b$ of the $\\alpha$-SQG equation around $\\bar\\Theta$ has an eigenvalue $\\lambda$ with $\\Re\\lambda>0$ and eigenfunction $W\\in U_{jn}$, and the nonlinear correction to $\\bar\\Theta+\\varepsilon\\Theta_{\\rm lin}+\\varepsilon^2\\Theta_{\\rm cor}$ stays of order $o(e^{\\Re\\lambda\\tau})$. The proof builds $\\bar\\Theta$ through a piecewise constant ansatz whose instability is decided by a discriminant $\\Delta(\\sigma)$, a fixed-point regularization, a spectral transfer to self-similar coordinates with a skew-adjoint plus compact decomposition needed for $\\alpha=1$, and weighted $Y^m$ energy estimates with an inductive ordering of polar-coordinate derivatives.","core_discovery":"Theorem 1.1 asserts: for every $0\\le\\alpha\\le1$, $s\\ge0$, $1\\le p\\le\\infty$ with $s<\\alpha+2/p$, there exist $T>0$ and a forcing term $f\\in L^1([0,T], W^{s,p}\\cap \\dot H^{(\\alpha-2)/2})$ such that the forced $\\alpha$-SQG equation has uncountably many solutions $\\theta_\\varepsilon\\in L^\\infty([0,T], W^{s,p}\\cap \\dot H^{(\\alpha-2)/2})$ with $\\theta_\\varepsilon(0)=0$. The proof exhibits a background vortex $\\bar\\Theta$, an unstable mode $W$, and a correction $\\Theta_{\\rm cor}$ so that $\\theta_\\varepsilon$ is built from $\\bar\\Theta+\\varepsilon\\Theta_{\\rm lin}+\\varepsilon^2\\Theta_{\\rm cor}$; different values of $\\varepsilon$ give different solutions. The same unstable data, run backward in time, produce the corollary of global smooth unforced solutions converging to the vortex, and the regularity is high enough that the non-unique solutions still satisfy the Hamiltonian identity and the renormalization property.","pith_inferences":["The discriminant calculation in Proposition 4.1 is carried out for $0\\le\\alpha<2$, so if the regularization and spectral steps could be extended beyond $\\alpha=1$, the non-uniqueness theorem itself would likely extend into the hyperdissipative range $1<\\alpha<2$.","The physical-time convergence rate of the global solutions of Corollary 1.3 is a power law, since $\\tau\\sim\\log t$ turns $e^{\\Re\\lambda\\tau}$ into $t^{\\Re\\lambda/(ab)}$; a numerical experiment tracking perturbations backward should measure this exponent, providing a direct test of the instability construction.","The inductive polar-coordinate energy scheme is organized so that each derivative estimate uses only lower-index estimates; this suggests the same spaces $Y^m$ could be used to prove instability-driven non-uniqueness for other active scalars whose Cartesian energy estimates fail near the origin."],"forward_implications":["Non-uniqueness with forcing holds simultaneously in every Sobolev space below the critical line, and by Corollary 1.1 also in the supercritical H\\\"older scale $\\Lambda^{-1}\\theta\\in C^\\gamma$ for $\\gamma<1+\\alpha$.","The non-unique solutions satisfy the Hamiltonian identity and the renormalization property, so neither conservation law singles out a unique flow in the forced supercritical regime.","For every $n\\ge2$ there exist smooth compactly supported $n$-fold symmetric vortices that are nonlinearly unstable for the unforced equation, with solutions exiting any small $H^m$ neighborhood of the vortex for $m>\\alpha+1$.","Reversing time on the unstable manifold gives global smooth unforced $\\alpha$-SQG solutions that are $n$-fold symmetric, non-stationary, neither rotating nor traveling, and converge to $\\bar\\Theta$ in $H^m$ as $t\\to\\infty$."],"supporting_citations":[{"why":"Supplies the forced 2D Euler non-uniqueness theorem that this paper generalizes to $0\\le\\alpha\\le1$.","marker":"[127,128]"},{"why":"Provides the detailed book exposition of Vishik's unstable-vortex strategy used as the template for the four-step proof.","marker":"[2]"},{"why":"The authors' simplified proof for forced 2D Euler, whose steps this paper adapts and extends to the $\\alpha$-SQG family.","marker":"[31]"},{"why":"Introduces the self-similar instability mechanism that converts linear exponential growth into non-uniqueness.","marker":"[97]"},{"why":"Gives the local well-posedness theory for $\\alpha$-SQG in borderline Sobolev spaces used to run the iterative scheme in $H^{m+2}$.","marker":"[35]"},{"why":"Provides the linear-to-nonlinear instability program whose energy-estimate structure the nonlinear step follows.","marker":"[82]"}],"fun_headline_variants":["Unstable vortices shatter uniqueness in forced SQG","Forced SQG: sharp non-uniqueness below critical smoothness","Uncountably many forced SQG flows from one force","Unstable vortices yield global smooth SQG solutions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the SQG regularization fixed point (Proposition 5.2, whose proof is omitted as analogous) actually exists with the stated bounds, together with the $H^m$ time-reversal regularity that depends on the same bootstrap; if that fixed point fails, the smooth unstable vortex used in every theorem disappears.","fun_headline_variants_meta":{"raw":{"variants":["Unstable vortices shatter uniqueness in forced SQG","Forced SQG: sharp non-uniqueness below critical smoothness","Uncountably many forced SQG flows from one force","Unstable vortices yield global smooth SQG solutions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000766,"raw_usage":{"total_tokens":3375,"prompt_tokens":900,"completion_tokens":2475,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":516,"completion_tokens_details":{"reasoning_tokens":2406}},"tokens_in":516,"tokens_out":2475,"duration_ms":18449,"temperature":1.0,"reasoning_tokens":2406,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T18:41:59.360078+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the discriminant $\\Delta(\\sigma)$ of Lemma 4.6 for $\\alpha=1$ and $n=2$ by numerical quadrature of the kernels $I_{1,1}$ and $I_{2,1}$ on $\\sigma\\in(1/2,1)$; Proposition 4.1 predicts $\\Delta(1)=\\Delta'(1)=0$ and $\\Delta''(1)<0$, so $\\Delta<0$ for $\\sigma$ just below 1, and finding $\\Delta\\ge0$ on a sequence tending to 1 would refute the claimed unstable vortex, as would an explicit failure of the contraction in Proposition 5.2 for $\\alpha=1$.","supporting_citations":[{"cited_title":"Jia and V","cited_arxiv_id":null,"evidence_quote":"Introduces the self-similar instability mechanism that converts linear exponential growth into non-uniqueness."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the linear-to-nonlinear instability program whose energy-estimate structure the nonlinear step follows."}],"review_version":1}