{"id":"551e69a1-e730-4d9f-af69-135a9ba03b01","arxiv_id":"2507.15739","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves superlinear vorticity-gradient growth for an open set of smooth torus data and for smooth compactly supported plane data near the Lamb dipole.","lead":"This paper proves that solutions of the two-dimensional Euler equations can have vorticity gradients that grow faster than any linear rate in time, even in domains with no boundary. It is the first such result for an open set of smooth initial data on the torus and the first superlinear growth result for smooth compactly supported vorticity in the whole plane.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Abstract overstates the torus theorem by omitting the no-smaller-period condition (1.6), which the proof of Proposition 2.1 uses essentially to prevent the approximate translation from switching branches.","rationale":"The central construction for the torus is carefully executed; Lemmas 2.2, 2.5, and the moving-frame reduction are sound. The weakest point in the paper's main advertised claim is not an internal proof error in the torus theorem but a mismatch between the abstract and Theorem 1.3: the no-smaller-period assumption (1.6) is essential to the proof of Proposition 2.1, where it guarantees that the discrete label s(t) cannot jump between the finitely many solutions of the phase-locking equation. The example cos(2x1)+cos(2x2) shows that without (1.6) the set S contains nonzero translation periods and the key constant c(omega*) vanishes, so the proof mechanism stops. The authors acknowledge this by imposing (1.6) and noting it might be removable, but the abstract does not carry the restriction. This is a concrete, checkable limitation rather than a speculative one. The plane result additionally depends on the quoted Abe-Choi stability theorem; that is a legitimate black-box concern the reader raised, but it is a dependency on a published result rather than an internal inconsistency, and I do not see a flaw in how Proposition 3.3 uses it. The reader's CONDITIONAL verdict is appropriate; the abstract should be corrected or the theorem proved in the claimed generality. I agree partially with the reader's weakest_assumption: the (1.6) omission is the more load-bearing of the two issues for the paper's headline claim.","tokens_in":25898,"tokens_out":36839,"duration_ms":409646,"concrete_test":"Take the steady state omega*(x)=cos(2x1)+cos(2x2), which has saddle points and is a periodic perturbation of the example family but violates (1.6). Try to run the proof of Proposition 2.1: compute the admissible set S from the two Fourier modes k(1)=(2,0), k(2)=(0,2); S contains (pi,0),(0,pi),(pi,pi). Show that for a solution whose translation a(t) winds around the torus, the nearest-element index s(t) must jump, so the estimate (2.1) cannot hold for a single C^1 p(t). If the jump cannot be ruled out, revise the abstract to include (1.6) or extend the proof to cover periodic states.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The advertised torus result in the abstract is strictly stronger than Theorem 1.3: the abstract drops the no-smaller-period hypothesis (1.6). This assumption is not decorative. In the proof of Proposition 2.1, p(t) is chosen among the finitely many solutions of Kp(t)=b(t) (2.5); these solutions differ by elements of S in (2.7). Orbital stability only gives a(t) with a(t)-p(t) close to S, and the proof defines s(t) as the nearest element in S. The step forcing s(t) identically 0 uses (1.6) to obtain c(omega*) := min_{s in S\\{0}} ||omega* - omega*(.-s)||_{L2} > 0, and then applies (2.14) to rule out jumps of s(t). If (1.6) fails, as for omega*(x)=cos(2x1)+cos(2x2), S contains nonzero periods, c(omega*)=0, and the argument that a single C^1 approximate translation p(t) works for all times has no justification; p(t) could switch between branches. The claim 'whenever we have a steady state orbitally stable up to translation and has a saddle point' is therefore unsupported without adding (1.6).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves two superlinear vorticity-gradient growth results for the 2D Euler equation in domains without boundary. On the torus, assuming a C^1 steady state is orbitally stable up to translation, has a saddle point, and has no nontrivial smaller period (condition (1.6)), the authors construct smooth initial data arbitrarily close in L^2 such that a whole L^infinity-neighborhood of that data produces solutions satisfying the bound (1.7), which implies superlinear growth (1.8). The proof introduces a differentiable approximate translation vector p(t) via Fourier phases, studies the equation in a moving frame, and uses a flux-through-a-parallelogram mechanism (Appendix A) inherited from Denisov. On the whole plane, the authors prove the first compactly supported smooth superlinear growth result by perturbing the Lamb dipole in the odd/positive symmetry class Xodd,+, relying on the orbital stability theorem of Abe and Choi as a black box. The torus and plane arguments are independent.","tokens_in":26167,"tokens_out":20836,"duration_ms":235289,"significance":"If correct, this is a notable advance: it removes symmetry assumptions on the initial data for small-scale creation on the torus, producing an L^infinity-open set of smooth data with superlinear gradient growth, and it gives the first superlinear growth for smooth compactly supported vorticity in the plane. The proof strategy is robust and conceptual, and the Appendix usefully isolates the flux-to-growth mechanism. The results are conditional on the quoted orbital stability theorems, which is standard for this line of work. The main weakness is that the abstract overstates the torus theorem by omitting the no-smaller-period condition (1.6), a hypothesis that is used essentially in the proof.","major_comments":[{"comment":"The abstract states the torus result for every steady state that is orbitally stable up to translation and has a saddle point, but Theorem 1.3 and Proposition 2.1 additionally require the no-smaller-period condition (1.6). This is not a harmless omission: in the proof of Proposition 2.1, after equations (2.10)-(2.11), the argument needs c(omega*) = min_{0 neq s in S} ||omega* - omega*(.-s)||_{L2} > 0, which follows from (1.6), to conclude from (2.14) that s(t) cannot jump and hence s(t) = 0 for all t. For a steady state such as omega*(x) = cos(2x1)+cos(2x2), the set S contains nonzero periods, c(omega*) = 0, and the proof gives no justification that a single C^1 approximate translation p(t) works for all times. The abstract must either include (1.6) or the authors must prove the stronger statement they advertise.","section":"Abstract vs. Theorem 1.3 and Proposition 2.1"}],"minor_comments":[{"comment":"The text refers to 'Multiplying -i to (2.30)' but equation (2.30) does not exist; the reference should be to (2.18).","section":"Section 2.2, proof of (2.2)"},{"comment":"The displayed inequality ||omega~_{p,epsilon}||_{L1} <= C ||omega~_{p,epsilon}||_{L2}^{1/2} (|supp|)^{1/2} is not valid in general; by Cauchy-Schwarz the correct bound is ||f||_{L1} <= |supp f|^{1/2} ||f||_{L2}, which is stronger and suffices for the argument.","section":"Section 3.2, equation (3.24)"},{"comment":"In the verification of Condition 2, the second curve is labeled 'C1' again; it should be C2 = {|x1+1| <= eta, x2 = -eta/2}.","section":"Section 3.3, proof of Theorem 1.5"},{"comment":"In the definition of g, the third case should be -3 for x1 <= -3, not 'x1 <= 3' as written.","section":"Footnote 9"},{"comment":"The definition of p(0) refers to 'a(0) given by Definition 1.1', but Definition 1.1 does not specify a unique a(t); the proof should clarify that a particular a(t) is fixed from the orbital stability statement and p(0) is measured against that choice.","section":"Definition 2.3"},{"comment":"The assertion that the level sets {mu(t,.) = 1} and {mu(t,.) = 2} always have a connected component touching both Gamma1 and Gamma3 is stated without proof. Since this is a load-bearing topological step, the authors should either give a proof or cite the precise argument in [4].","section":"Appendix A, Lemma A.1"}],"recommendation":"major_revision","confidential_remarks":"The central mechanism is credible and the paper is well written. The main obstacle is the mismatch between the abstract and the proved torus theorem: the no-smaller-period assumption (1.6) is essential to the proof of Proposition 2.1. The authors themselves note that removing it would require extra work, so the abstract should be corrected. The plane result is a meaningful conditional contribution; the reliance on Abe-Choi is appropriate. I see no novelty or citation-practice concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my take on Jeong-Yao-Zhou, arXiv:2507.15739. It proves the first superlinear vorticity gradient growth for an open set of smooth data on the torus without symmetry assumptions, and the first superlinear growth for smooth compactly supported vorticity in the plane. Both are genuine advances, not repackaged earlier work. The torus result assumes orbital stability up to translation plus a saddle point plus a no-smaller-period condition; the plane result uses the published Abe-Choi stability of the Lamb dipole. The proof mechanism — the approximate translation vector p(t), via Fourier phase tracking on the torus and an implicit weighted integral on the plane — is new and worked out in detail. Appendix A cleanly abstracts Denisov's flux argument, and the integral bound on the inverse gradient is a nice strengthening. I find the paper strong and worth serious engagement.\n\nNow the soft spots. The stress-test note is right: the abstract overstates the torus theorem. It says \"whenever we have a steady state orbitally stable up to a translation and has a saddle point,\" but Theorem 1.3 and Proposition 2.1 both impose (1.6), no smaller period. That condition is used essentially. It gives c(omega*) > 0, which prevents the discrete translation s(t) from jumping between branches in Proposition 2.1. Without (1.6), the proof as written does not justify a single C^1 approximate translation p(t) for all times. This is not a cosmetic mismatch; the advertised claim is stronger than what is proved. It is fixable — the authors even say they expect (1.6) can be removed — but as posted, it needs correction. A referee should ask for the abstract to carry the hypothesis or the theorem to be extended.\n\nThe R^2 result leans entirely on Abe-Choi's orbital stability as a black box. That is a legitimate published theorem, and the paper is transparent about inheriting its symmetry class Xodd,+. This is a boundary of the result, not a flaw. The citation pattern is clean; the only self-citation is background on the Lamb dipole. No circularity.\n\nWho should read this: people working on small-scale formation in 2D Euler, and anyone interested in what stability plus a saddle point can buy without symmetry. The paper deserves a serious referee. I would recommend minor revision: align the abstract with Theorem 1.3 by adding (1.6), or extend the proof to remove it.","headline":"Real progress: first symmetry-free open-set superlinear gradient growth on the torus and first superlinear growth in the plane, but the abstract overclaims the torus theorem by dropping the no-smaller-period condition (1.6).","tokens_in":26658,"tokens_out":2370,"would_cite":true,"duration_ms":29379,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q31","76B03","35B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that superlinear growth of the vorticity gradient—a standard signature of small-scale formation in 2D ideal flow—occurs robustly for an open set of smooth initial data near any orbitally stable, saddle-point steady state…","keywords":["2D Euler equations","vorticity gradient growth","small scale creation","orbital stability up to translation","Lamb dipole","superlinear growth","open set of initial data","torus"],"falsifier":"Fix $\\omega^*=\\cos x_1+\\cos x_2$ on $\\mathbb{T}^2$, construct $\\tilde\\omega_0$ as in Theorem 1.3, and numerically integrate 2D Euler from a $C^1$ initial datum within the stated $L^\\infty$ neighborhood, measuring $I(T)=\\int_0^T \\|\\nabla\\omega(t)\\|_{L^\\infty}^{-1}\\,dt$. If $I(T)$ fails to stay bounded, or if the limsup of $\\|\\nabla\\omega(t)\\|/(t\\log t)$ remains finite, Theorem 1.3 is false. Equivalently, exhibiting a steady state satisfying all hypotheses—including (1.6)—for which every nearby $C^1$ datum has the integral divergent would refute the claimed mechanism.","tokens_in":25700,"feed_emoji":"🌀","tokens_out":6722,"duration_ms":69799,"temperature":0.7,"pith_summary":"The paper proves that superlinear growth of the vorticity gradient, the accepted signature of small-scale formation in two-dimensional ideal fluids, occurs robustly near a large class of stable coherent states, without any symmetry assumptions on the data. In the torus, any steady state that is orbitally stable up to translation, has a saddle point, and has no smaller period serves as a seed: the authors construct smooth initial data arbitrarily close in $L^2$ to it such that every $C^1$ initial datum in a small $L^\\infty$ neighborhood has $\\int_0^\\infty \\|\\nabla\\omega\\|_{L^\\infty}^{-1}\\,dt$ finite, so $\\|\\nabla\\omega\\|_{L^\\infty}$ outgrows $t\\log t$ along a sequence of times. In the plane, the same conclusion is reached for smooth, compactly supported vorticity near the Lamb–Chaplygin dipole, within the odd symmetry class where the dipole is known to be stable; this is the first superlinear growth result of any kind for smooth compactly supported planar vorticity. A sympathetic reader would care because it converts a phenomenon previously obtained only through fragile symmetry assumptions into a stable, open-set phenomenon.","feed_headline":"Superlinear gradient growth holds without symmetry in 2D Euler","feed_subtitle":"Torus near stable saddle states; first such result in the plane near the Lamb dipole.","key_machinery":"The central object is an approximate translation vector $p(t)$, a $C^1$ curve with arbitrarily small speed that tracks the unknown translation $a(t)$ in the orbital-stability definition closely enough that the solution stays close to the steady state in the moving frame $x\\mapsto x+p(t)$. On the torus $p(t)$ is built from the phases of two non-vanishing Fourier modes of $\\omega^*$, solving a $2\\times 2$ linear system $Kp(t)=b(t)$; in the plane it is defined implicitly by $\\int_{\\mathbb{R}^2_+}\\bar\\omega(t,x)g(x_1-p(t))\\,dx=0$ with a bounded odd cutoff $g$. This $p(t)$ converts the purely $L^2$ closeness of orbital stability into a uniform velocity bound $\\|v(t)-u^*\\|_{L^\\infty}\\le C\\sqrt{\\varepsilon}$ for the moving-frame velocity $v=\\nabla^\\perp\\Delta^{-1}\\rho-\\dot p$, which preserves the strict inward/outward flux of $u^*$ across the sides of a small parallelogram (torus) or square (plane) around the saddle point. With signed flux and initial level sets of different values crossing the domain, a transport lemma (after Denisov) forces the gradient integral to be finite.","core_discovery":"On the torus, for a steady state $\\omega^*$ that is orbitally stable up to translation and whose flow has a saddle point, the paper constructs a smooth perturbation $\\tilde\\omega_0$ at arbitrarily small $L^2$ distance from $\\omega^*$ such that every $C^1$ initial datum $\\omega_0$ with $\\|\\omega_0-\\tilde\\omega_0\\|_{L^\\infty}$ small satisfies $\\int_0^\\infty \\|\\nabla\\omega(t)\\|_{L^\\infty}^{-1}\\,dt < C_0$, which implies $\\limsup_{t\\to\\infty} \\|\\nabla\\omega(t)\\|_{L^\\infty}/(t\\log t)=\\infty$. An explicit family of examples is $\\omega^*_{\\alpha,\\beta}=\\alpha\\cos x_1+\\beta\\cos x_2$, whose orbital stability was previously established. In $\\mathbb{R}^2$, the analogous statement holds for smooth compactly supported initial data near the Lamb dipole within the symmetry class $X^{\\mathrm{odd},+}$; the construction uses the dipole's two saddle points in the co-moving frame.","pith_inferences":["A plausible reading is that hyperbolic stagnation points in stably moving coherent structures act as universal small-scale generators in 2D ideal flow; the paper's open-set statement suggests this persists under generic, symmetry-free perturbations rather than being a measure-zero phenomenon.","For rotating equilibria such as Kirchhoff ellipses, the same mechanism should apply once a stability statement with propagating support control is available; the paper identifies exactly that missing ingredient.","Testable extension: the plane construction suggests a concrete quantitative prediction—near the Lamb dipole, $\\|\\nabla\\omega\\|_{L^\\infty}$ should grow at least like $t/(\\log t)$ along an explicitly computable subsequence; a high-resolution numerical check could validate the rate.","Because the open set is in $L^\\infty$ (and implicitly $C^1$), the constructed $\\tilde\\omega_0$ itself may be approximated by piecewise constant or vortex-blob data, suggesting a route to experimental or numerical verification of open-set small-scale creation."],"forward_implications":["On $\\mathbb{T}^2$, any orbitally stable-up-to-translation steady state with a saddle point and no smaller period automatically seeds an open set of $C^1$ data with $\\int_0^\\infty \\|\\nabla\\omega\\|_{L^\\infty}^{-1}\\,dt<\\infty$, hence superlinear gradient growth; no parity or rotational symmetry is needed.","In $\\mathbb{R}^2$, smooth compactly supported vorticity can now exhibit gradient growth faster than linear, closing the gap left by earlier linear-growth results near the Lamb dipole.","The time-integral bound is stronger than a limsup statement: it implies the same superlinear growth with a quantitative integrability guarantee, and yields the earlier averaged statement $\\frac1T\\int_0^T\\|\\nabla\\omega\\|\\,dt\\to\\infty$ as a corollary.","The proof scheme is transferable: any traveling or uniformly rotating relative equilibrium that is orbitally stable (up to translation or rotation) and has a saddle in its moving frame is a candidate for the same conclusion.","Condition (1.6) is technical; the authors expect it can be removed, which would widen the torus theorem to all orbitally stable steady states with saddles."],"supporting_citations":[{"why":"Supplies the orbital stability of the Lamb dipole in $X^{\\mathrm{odd},+}$ up to translation, which is the black-box premise for the plane result (Proposition 3.2).","marker":"[1]"},{"why":"Provides the signed-flux superlinear-growth argument (Lemma A.1) and the earlier symmetry-dependent torus example that this paper generalizes to an open set.","marker":"[4]"},{"why":"Establishes orbital stability up to translation for the sinusoidal steady states $\\omega^*_{\\alpha,\\beta}=\\alpha\\cos x_1+\\beta\\cos x_2$, giving the concrete family satisfying the torus theorem's hypotheses.","marker":"[18]"},{"why":"Adds quantitative $\\delta(\\varepsilon)$ bounds for the stability of the sinusoidal steady states, used to justify the example family in Theorem 1.3.","marker":"[7]"},{"why":"Provides the prior linear-growth result near the Lamb dipole and the review of Lamb dipole properties (moving frame, saddle points) that the plane construction builds on.","marker":"[3]"}],"fun_headline_variants":["Superlinear vorticity growth without symmetry in 2D Euler","First symmetry-free superlinear growth for 2D Euler","Open-set superlinear growth for 2D Euler vorticity","Lamb dipole drives superlinear growth in the plane","Torus saddle steady states give superlinear gradient growth"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes as a black box that the chosen steady state really is orbitally stable up to translation (for the plane, that the Lamb dipole is stable in $X^{\\mathrm{odd},+}$), and on the torus it additionally assumes the steady state has no smaller period; if those stability or non-degeneracy facts fail, the construction has no ground to stand on.","fun_headline_variants_meta":{"raw":{"variants":["Superlinear vorticity growth without symmetry in 2D Euler","First symmetry-free superlinear growth for 2D Euler","Open-set superlinear growth for 2D Euler vorticity","Lamb dipole drives superlinear growth in the plane","Torus saddle steady states give superlinear gradient growth"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000515,"raw_usage":{"total_tokens":2494,"prompt_tokens":935,"completion_tokens":1559,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":551,"completion_tokens_details":{"reasoning_tokens":1478}},"tokens_in":551,"tokens_out":1559,"duration_ms":11803,"temperature":1.0,"reasoning_tokens":1478,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:25:47.790627+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix $\\omega^*=\\cos x_1+\\cos x_2$ on $\\mathbb{T}^2$, construct $\\tilde\\omega_0$ as in Theorem 1.3, and numerically integrate 2D Euler from a $C^1$ initial datum within the stated $L^\\infty$ neighborhood, measuring $I(T)=\\int_0^T \\|\\nabla\\omega(t)\\|_{L^\\infty}^{-1}\\,dt$. If $I(T)$ fails to stay bounded, or if the limsup of $\\|\\nabla\\omega(t)\\|/(t\\log t)$ remains finite, Theorem 1.3 is false. Equivalently, exhibiting a steady state satisfying all hypotheses—including (1.6)—for which every nearby $C^1$ datum has the integral divergent would refute the claimed mechanism.","supporting_citations":[{"cited_title":"Abe and K","cited_arxiv_id":null,"evidence_quote":"Supplies the orbital stability of the Lamb dipole in $X^{\\mathrm{odd},+}$ up to translation, which is the black-box premise for the plane result (Proposition 3.2)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the signed-flux superlinear-growth argument (Lemma A.1) and the earlier symmetry-dependent torus example that this paper generalizes to an open set."},{"cited_title":"W ang and B","cited_arxiv_id":null,"evidence_quote":"Establishes orbital stability up to translation for the sinusoidal steady states $\\omega^*_{\\alpha,\\beta}=\\alpha\\cos x_1+\\beta\\cos x_2$, giving the concrete family satisfying the torus theorem's hypotheses."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Adds quantitative $\\delta(\\varepsilon)$ bounds for the stability of the sinusoidal steady states, used to justify the example family in Theorem 1.3."},{"cited_title":"Choi and I.-J","cited_arxiv_id":null,"evidence_quote":"Provides the prior linear-growth result near the Lamb dipole and the review of Lamb dipole properties (moving frame, saddle points) that the plane construction builds on."}],"review_version":1}