{"id":"aacffe74-939e-4e74-be9b-aca3d4932a4d","arxiv_id":"2506.02800","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Near the first excited state -sin 2θ of the De Gregorio model, some perturbations are unstable while a smaller even-mode class decays exponentially.","lead":"This paper studies the De Gregorio model, a one-dimensional stand-in for 3D fluid blow-up, near its first excited steady state -sin 2θ. It proves that small perturbations of this state can either grow or decay depending on the initial shape of the perturbation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central instability claim survives scrutiny; the load-bearing weakness is Theorem 1.4's stability proof, where inequality (6.11) has incorrect powers and the key decay estimate (6.5) is imported rather than derived.","rationale":"The reader's conditional verdict is reasonable, and my independent read agrees that the instability half is the stronger part. The reader's flagged concern about (6.5) is real but not fatal: although the paper cites [12] instead of deriving the decay inequality, the inequality follows immediately from the paper's own diagonal formula (1.21) once one notes that the even-index coefficients c_{2j}=d_{2j}-d_{2j+2} are all ≤ -3/8. Thus the stability mechanism is recoverable, but the paper should supply that one-line argument. The additional issue I found is in (6.11): the powers in the inequality appear to be dimensionally inconsistent with the preceding estimates, and the printed form does not yield the claimed exponential decay. Since this affects the proof of one of the two advertised conclusions but not the central instability theorems, the appropriate verdict remains conditional rather than accept or reject. The omitted proofs of Theorem 1.1′ and Lemma 5.2 are auxiliary and do not change the assessment.","tokens_in":28244,"tokens_out":45332,"duration_ms":411161,"concrete_test":"Recompute the Gronwall step of Theorem 1.4 with X=⟨η,η⟩_ρ: verify whether (6.11) should read dX/dt ≤ -(3/4)X + C X^{3/2}, and check that the claimed rate e^{-3t/8} is obtained for X(0) small. Also verify (6.5) directly from (1.21) on even indices by showing d_{2j}-d_{2j+2}≤-3/8 for all j≥1; if either check fails, Theorem 1.4 is not proved as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's advertised central contribution is the instability of the excited state; Theorems 1.2 and 1.3 appear internally consistent: the second-order ODE inequality (4.23) follows from the diagonalized form (1.21) plus Lemma 4.1, and the comparison argument with J1, J2 is valid. The soft spot is the stability half, Theorem 1.4. Inequality (6.5) is cited from [12] without proof for the linearized operator around -sin 2θ on the even-tilded subspace. It can be supplied from (1.21) and the bound c_{2j}=d_{2j}-d_{2j+2}≤-3/8 (from (4.10)), so this is a presentation gap rather than a fatal error. More serious is (6.11): after estimating the nonlinear term by C∥ρ^{1/2}∂θη∥_{L2}^3 = C⟨η,η⟩_ρ^{3/2}, the paper concludes d⟨η,η⟩_ρ/dt ≤ -(3/4)⟨η,η⟩_ρ^2 + C⟨η,η⟩_ρ^3. The correct form from (6.6) is d⟨η,η⟩_ρ/dt ≤ -(3/4)⟨η,η⟩_ρ + C⟨η,η⟩_ρ^{3/2}. With the printed powers, the claimed exponential decay ∥η∥_HDW ≲ e^{-3t/8}∥η0∥_HDW does not follow; with the corrected inequality it follows for sufficiently small initial data. Theorem 1.4's proof must be repaired, and the auxiliary existence statements Theorem 1.1′ and Lemma 5.2 are only sketched.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the De Gregorio modification of the Constantin-Lax-Majda model on the torus, focusing on the first excited state -sin 2θ. It establishes linear instability (Theorem 1.2) for odd perturbations satisfying an initial sign condition, nonlinear instability in a Lipschitz sense (Theorem 1.3), and nonlinear stability with exponential decay for initial data supported on even-index tilded basis functions (Theorem 1.4). The proofs rely on a weighted Hilbert space HDW, an infinite ODE system for Fourier coefficients, a second-order differential inequality with explicit spectral constants, and comparison arguments.","tokens_in":28573,"tokens_out":15127,"duration_ms":116389,"significance":"If the results hold, this is a significant step beyond the ground-state analysis of the De Gregorio model, addressing the sign-indefinite linearized operator around an excited state. The instability proof is detailed and self-contained: it provides explicit algebraic constants (Lemmas 4.1 and 7.1), parameter-free derivations, and numerical verification of the spectral bounds. The nonlinear instability mechanism and the complementary stability statement give a nuanced picture of solution behavior near -sin 2θ. However, the stability half (Theorem 1.4) currently relies on an incorrect differential inequality and an imported decay estimate, so its validity is not yet established in the manuscript.","major_comments":[{"comment":"The differential inequality (6.11) has incorrect powers. The estimates (6.7)-(6.10) bound each nonlinear term by a constant times ∥ρ^{1/2}∂θη∥_{L2}^3 = ⟨η,η⟩_ρ^{3/2}. Substituting these into (6.6) yields d⟨η,η⟩_ρ/dt ≤ -(3/4)⟨η,η⟩_ρ + C⟨η,η⟩_ρ^{3/2}, not the printed version -(3/4)⟨η,η⟩_ρ^2 + C⟨η,η⟩_ρ^3. With the printed powers, the claimed exponential decay ∥η∥_{HDW} ≲ e^{-3t/8}∥η0∥_{HDW} does not follow; with the corrected inequality it does follow for sufficiently small initial data. The proof of Theorem 1.4 must be repaired.","section":"Section 6, Eq. (6.11)"},{"comment":"The key decay inequality ⟨−Lη,η⟩_ρ ≤ -(3/8)⟨η,η⟩_ρ for the linearized operator around -sin 2θ on the even-index tilded subspace is cited from [12] without proof or verification. This inequality is the sole source of exponential decay in Theorem 1.4, and the cited work addresses the ground state -sin θ. The authors should either derive (6.5) from (1.21) and (4.10) (where d_{2j}-d_{2j+2} ≤ -3/8 is immediate) or state and prove the transfer to the present setting. As written, the stability half of the paper rests on an unverified external input.","section":"Section 6, Eq. (6.5)"}],"minor_comments":[{"comment":"The result is stated without proof, with only a note that the proof is analogous to that of Theorem 1.1. Since it is not used in the main arguments, please either provide a proof in an appendix or move it to a remark.","section":"Section 1, Theorem 1.1'"},{"comment":"Lemma 5.2 is also stated without proof. If it is intended as an auxiliary result, please prove it or mark it as a remark; otherwise remove it to avoid unsupported claims.","section":"Section 5, Lemma 5.2"},{"comment":"There are typos in the definition of H^m(T): 'f or all' should be 'for all'.","section":"Section 1, definition of H^m"},{"comment":"'Asuume' and 'exits' should be 'Assume' and 'exists'.","section":"Section 1, Theorem 1.1'"},{"comment":"In the displayed expansion of S_n, the term (-d_6+d_4)^2 appears with η_6^2; from the definition of f_2 it should be η_4^2. Also, the first two terms of the expansion appear to be duplicated in the displayed formula.","section":"Section 4, Eq. (4.15)"},{"comment":"The numerical figures (a)-(e) are referenced but not actually embedded in the text; please include them or state that they are available as supplementary material.","section":"Section 7, Appendix B"},{"comment":"The lower bound in (1.20) is ambiguous due to missing parentheses; it should read a_k^2 ≥ (11/18 - √λ1)/(√λ1 + d_{k+2} - d_k) a_1^2.","section":"Remark 1.3, Eq. (1.20)"}],"recommendation":"major_revision","confidential_remarks":"The instability part (Theorems 1.2 and 1.3) is well supported, with explicit constants and numerical checks. The stability part (Theorem 1.4) contains an incorrect inequality and an imported decay estimate; I am confident the authors can fix these, but the paper should not be accepted in the current form. The unproved auxiliary statements should also be addressed. The fit with the journal's scope is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core result here is solid: the paper gives the first analytic treatment of stability around -sin 2θ for the De Gregorio model, and the main instability theorem (Theorem 1.2) checks out. The second-order ODE inequality and the uniform spectral bounds on the quadratic form f_k are genuinely new, and the explicit constants in Lemma 7.1 are proven with real computation, not numerical hand-waving. The nonlinear transfer in Theorem 1.3 follows the standard scaling/contradiction route and looks internally consistent. This is a legitimate advance for the subfield of 1D fluid-inspired models, and it addresses a gap explicitly left open in [11,12].\n\nThe soft spots are real but concentrated. Theorem 1.4's proof of nonlinear stability contains a concrete error: starting from the inequality with 1/2 d/dt <η,η>_ρ ≤ -3/8 <η,η>_ρ + C<η,η>_ρ^{3/2}, the paper concludes d/dt <η,η>_ρ ≤ -(3/4)<η,η>_ρ^2 + C<η,η>_ρ^3. The correct form is -(3/4)<η,η>_ρ + C<η,η>_ρ^{3/2}. With the printed powers the claimed exponential decay does not follow; with the corrected inequality it follows for small initial data, so this is a repair, not a fatal flaw. Relatedly, the key decay estimate (6.5) is imported from [12] rather than derived for the excited state; the stress-test note is right that it can likely be supplied from (1.21) and the explicit coefficients, but the paper should show that work instead of citing it.\n\nTwo existence results, Theorem 1.1′ and Lemma 5.2, are only sketched with \"the proof is analogous,\" which is acceptable for a paper whose real content is spectral, but worth noting.\n\nBottom line: the central instability claim is sound and the paper deserves serious refereeing. The referee should insist on fixing Theorem 1.4 and either deriving or precisely citing (6.5), but the contribution is not in doubt.","headline":"The linear instability claim for the first excited state survives close reading, but Theorem 1.4's nonlinear stability proof has a repairable error in the differential inequality and relies on an imported decay estimate.","tokens_in":29106,"tokens_out":1299,"would_cite":true,"duration_ms":14449,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B10","35B35","35C10","35Q35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Around the first excited state of the De Gregorio model, the paper proves that one class of perturbations grows exponentially while another decays, so no uniform bound holds for all small data.","keywords":["De Gregorio model","Constantin-Lax-Majda model","excited states","linear instability","nonlinear instability","nonlinear stability","Hilbert transform","Fourier coefficients"],"falsifier":"Evaluate $\\langle -L\\eta,\\eta\\rangle_\\rho + \\frac{3}{8}\\langle \\eta,\\eta\\rangle_\\rho$ for each even-index tilded basis element $\\tilde e_{2k}^{(o)}$; the first negative value would refute the decay claim behind Theorem 1.4. An even simpler check is to solve the linear equation (1.8) from small even-tilded data and see whether $\\|\\eta(t)\\|_{H^{DW}}$ actually decays at the quoted rate.","tokens_in":28020,"feed_emoji":"🌊","tokens_out":9733,"duration_ms":81890,"temperature":0.7,"pith_summary":"This paper studies the De Gregorio modification of the Constantin-Lax-Majda model on the torus near the first excited steady state $-\\sin 2\\theta$, a one-dimensional vorticity model built to mimic aspects of the 3D Euler equations. The authors prove that stability near this state depends on the initial data: odd perturbations whose weighted energy satisfies $\\langle -L\\eta_0,\\eta_0\\rangle_\\rho\\ge 0$ are linearly unstable, with the perturbation norm trapped between two explicit exponentially growing curves. They transfer this result to the nonlinear equation, showing that for arbitrarily small smooth data one can force $\\|u(t)\\|_{L^2}$ past any linear growth bound $F(y)\\le Ky$, so the usual uniform stability estimate (1.19) cannot hold. For a different class of data consisting of even-index tilded modes, they prove global well-posedness and exponential decay of the perturbation in the weighted space at rate $3/8$. The paper matters because excited-state stability of the De Gregorio model was left open after ground-state results, and the same mechanism may transfer to other one-dimensional Euler-inspired models.","feed_headline":"De Gregorio first excited state is unstable for broad class of data","feed_subtitle":"Near -sin 2θ, small odd perturbations grow exponentially while even-index perturbations decay; initial data decides.","key_machinery":"The load-bearing object is the odd basis $\\tilde e_k^{(o)} = e_{k+2}^{(o)}/(k+2)-e_k^{(o)}/k$ in the weighted Hilbert space $H^{DW}$ with inner product $\\langle \\xi,\\eta\\rangle_\\rho = \\int_\\mathbb{T} \\rho\\,\\partial_\\theta\\xi\\,\\partial_\\theta\\eta\\,d\\theta$, with $\\rho = \\sin^2\\theta/(4\\pi)$; this basis diagonalizes the linearized operator's action up to a three-term coupling. Writing the linearized solution as $\\eta=\\sum_k \\tilde\\eta_k(t)\\tilde e_k^{(o)}$ turns the equation into the infinite ODE system $\\tilde\\eta_k' = -d_k\\tilde\\eta_{k-2}+(d_k-d_{k+2})\\tilde\\eta_k+d_{k+2}\\tilde\\eta_{k+2}$. The paper then differentiates the weighted energy $\\sum_k(\\tilde\\eta_k)^2$ twice and organizes the resulting expression into local quadratic forms $f_k$ in $(\\tilde\\eta_k,\\tilde\\eta_{k+2})$; Lemma 4.1 shows every $f_k$ is positive definite with eigenvalues uniformly trapped between $\\lambda_1$ and $\\lambda_2$, yielding the differential inequality $4\\lambda_1\\|\\eta\\|^2 < \\frac{d^2}{dt^2}\\|\\eta\\|^2 < 4\\lambda_2\\|\\eta\\|^2$. A comparison theorem for second-order ODEs converts that inequality into the exponential brackets of Theorem 1.2. The stability half is carried by a separate decay estimate for $\\langle -L\\eta,\\eta\\rangle_\\rho$ quoted from the earlier ground-state analysis.","core_discovery":"The central claim is that the first excited state is neither stable nor unstable in an unconditional sense. For the linearized equation around $-\\sin 2\\theta$, every nonzero odd initial datum with $\\langle -L\\eta_0,\\eta_0\\rangle_\\rho\\ge 0$ gives a solution whose $H^{DW}$ norm satisfies $J_1^{1/2}(t)<\\|\\eta(t)\\|_{H^{DW}}<J_2^{1/2}(t)$, with positive absolute constants $1/50<\\lambda_1<\\lambda_2<3/5$; in particular the norm grows at least like a constant times $e^{\\sqrt{\\lambda_1}t}$. The same instability survives in the nonlinear problem in the Lipschitz sense: for any $\\delta>0$, $K>0$, and $F(y)\\le Ky$, there is smooth initial data with $H^m$ norm below $\\delta$ whose solution exceeds $F(\\|u_0\\|_{H^m})$ in $L^2$ at some finite time, ruling out (1.19). On the other hand, for initial data $\\eta_0=\\sum_{k\\ge 1}a_{2k}\\tilde e_{2k}^{(o)}$ that is small in $H^{DW}$, the nonlinear problem is globally well-posed and decays as $\\|\\eta(t)\\|_{H^{DW}}\\lesssim e^{-3t/8}\\|\\eta_0\\|_{H^{DW}}$. Thus the same steady state supports both exponential growth and exponential decay, with the Fourier support and coefficient balance of the initial data selecting the regime.","pith_inferences":["The same second-order ODE plus positive-definite quadratic-form mechanism should adapt to excited states $-\\sin k\\theta$ with $k\\ge 3$, with the spectral constants changing with $k$; the sign pattern of the coefficients should continue to decide which modes grow.","The sign change in the coefficient $-d_{k+2}+d_k$ (positive at $k=1$, negative for $k\\ge 2$) suggests a threshold surface in initial-data coefficient space separating growth from decay; locating it numerically near $-\\sin 2\\theta$ would be a direct test of the dichotomy.","A self-contained proof of the decay inequality (6.5) on the even-index tilded subspace would determine whether the rate $3/8$ is sharp and would remove the stability claim's dependence on transferring ground-state spectral information."],"forward_implications":["If the paper's claims are right, the uniform bound (1.19) fails near $-\\sin 2\\theta$: for any $F(y)\\le Ky$, some arbitrarily small smooth data grow until $\\|u(t_K)\\|_{L^2}>F(\\|u_0\\|_{H^m})$.","The linearized instability is quantitative: nonzero odd data with $\\langle -L\\eta_0,\\eta_0\\rangle_\\rho\\ge 0$ have their $H^{DW}$ norm bracketed by two explicit exponentials with rates controlled by absolute constants in $(1/50,3/5)$.","Small even-index data of the form $\\eta_0=\\sum_k a_{2k}\\tilde e_{2k}^{(o)}$ yield a globally well-posed nonlinear flow with exponential decay at rate $3/8$, so the same steady state admits both growing and decaying regimes.","The positivity and uniform bounds on the quadratic forms $f_k$ are the structural reason the instability is robust: the sign-changing infinite system is squeezed into a uniform second-order differential inequality."],"supporting_citations":[{"why":"provides the tilded basis, the weighted Hilbert space $H^{DW}$, and the decay inequality (6.5) that powers Theorem 1.4.","marker":"[12]"},{"why":"first rigorous stability analysis around the ground state and explicitly left the excited-state cases open.","marker":"[11]"},{"why":"comparison theorem for second-order differential inequalities that converts the differential inequality (4.23) into the exponential bounds of Theorem 1.2.","marker":"[13]"},{"why":"nonlinear instability strategy via scaled perturbations and contradiction that Theorem 1.3 adapts.","marker":"[19]"},{"why":"introduced the one-parameter family containing the De Gregorio model and reported numerical convergence to ground states, the context against which excited-state behavior is compared.","marker":"[15]"}],"fun_headline_variants":["De Gregorio -sin2θ state: growth or decay set by data","Same De Gregorio state: stable for some data, unstable for others","Stability near -sin2θ in De Gregorio depends on initial data","Data selects: De Gregorio first excited state grows or decays","First excited state of De Gregorio model is conditionally unstable"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The stable half of the paper assumes the linearized decay inequality $\\langle -L\\eta,\\eta\\rangle_\\rho \\le -\\frac{3}{8}\\langle \\eta,\\eta\\rangle_\\rho$ on the even-index tilded subspace, an estimate quoted from the earlier ground-state analysis rather than proved here; if that estimate does not extend to the first excited state, the exponential stabilization result collapses.","fun_headline_variants_meta":{"raw":{"variants":["De Gregorio -sin2θ state: growth or decay set by data","Same De Gregorio state: stable for some data, unstable for others","Stability near -sin2θ in De Gregorio depends on initial data","Data selects: De Gregorio first excited state grows or decays","First excited state of De Gregorio model is conditionally unstable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00146,"raw_usage":{"total_tokens":5955,"prompt_tokens":1108,"completion_tokens":4847,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":724,"completion_tokens_details":{"reasoning_tokens":4753}},"tokens_in":724,"tokens_out":4847,"duration_ms":32392,"temperature":1.0,"reasoning_tokens":4753,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:17:28.460759+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate $\\langle -L\\eta,\\eta\\rangle_\\rho + \\frac{3}{8}\\langle \\eta,\\eta\\rangle_\\rho$ for each even-index tilded basis element $\\tilde e_{2k}^{(o)}$; the first negative value would refute the decay claim behind Theorem 1.4. An even simpler check is to solve the linear equation (1.8) from small even-tilded data and see whether $\\|\\eta(t)\\|_{H^{DW}}$ actually decays at the quoted rate.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the tilded basis, the weighted Hilbert space $H^{DW}$, and the decay inequality (6.5) that powers Theorem 1.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"first rigorous stability analysis around the ground state and explicitly left the excited-state cases open."},{"cited_title":"Matos Peixoto","cited_arxiv_id":null,"evidence_quote":"comparison theorem for second-order differential inequalities that converts the differential inequality (4.23) into the exponential bounds of Theorem 1.2."},{"cited_title":"Jiang, S","cited_arxiv_id":null,"evidence_quote":"nonlinear instability strategy via scaled perturbations and contradiction that Theorem 1.3 adapts."},{"cited_title":"Okamoto, T","cited_arxiv_id":null,"evidence_quote":"introduced the one-parameter family containing the De Gregorio model and reported numerical convergence to ground states, the context against which excited-state behavior is compared."}],"review_version":1}