{"id":"c8fcfdbe-c7e7-4078-beba-ecd33e1b4130","arxiv_id":"2504.16861","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For almost all Weber numbers below 4(2+sqrt(3)), small-amplitude circular vortex sheets with surface tension stay O(ε)-close to the circular equilibrium for times of order ε^{-(N+1)} for every integer N.","lead":"The paper proves that small perturbations of a circular fluid interface, when both surface tension and a velocity jump are present, remain small for extremely long times, confirming a 1991 physics conjecture. The proof rewrites the vortex-sheet equations in Hamiltonian form and applies a high-powered normal-form analysis to control the nonlinear resonances.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's existence assertion depends on an outsourced local well-posedness result (Remark 7.11); without a verified transfer to the singular-integral system (1.9), the bootstrap has no solution to propagate.","rationale":"I read the paper's central claim as Theorem 1.1: almost-global existence for small data near circular vortex sheets. The proof is a long reduction: Hamiltonian formulation, paralinearization (Theorem 4.2), complex coordinates, reduction to constant coefficients (Proposition 6.1), Darboux symplectic correction, Hamiltonian Birkhoff normal form, and an energy estimate. The forward parts are detailed and plausible; I did not find an internal inconsistency in the non-resonance argument, including the special treatment of the (3,5) resonance, and the SAP Hamiltonian mechanism for excluding resonant growth is coherent. The single load-bearing gap is the local well-posedness theory. The reader identified exactly this: the bootstrap in Section 7.3 requires a solution that exists, and the manuscript does not prove local existence for (1.9); it delegates to [25,3,53]. Because Theorem 1.1 asserts existence and uniqueness, an a priori estimate alone is insufficient. This is a correctable gap rather than a demonstrated counterexample, so I concur with the CONDITIONAL verdict and do not recommend changing it. My proposed check is the natural one: a self-contained verification that [25]'s argument transfers to the singular-integral KP system with the stated functional setting.","tokens_in":84295,"tokens_out":15371,"duration_ms":154150,"concrete_test":"Write out a complete local well-posedness proof for (1.9) in H^{s+1/4}_0 × Hdot^{s-1/4}, following the scheme of [25] and verifying each hypothesis: confirm that the principal symbol in Theorem 4.2 admits a positive-definite symmetrizer, that the smoothing remainders satisfy the tame estimates of Definition 3.8, and that the time-derivative estimates of Lemma 7.12 close without extra derivatives. A decisive computational check is to run the standard iterative scheme for smooth data and see whether the energy inequality d/dt ||(η,ψ)||_s^2 ≤ C(||(η,ψ)||_{s0}) ||(η,ψ)||_s^2 closes with the same derivative count. If the transfer from [25] fails—for instance if closing the energy requires ψ ∈ H^{s+1/4} or loses derivatives through the singular-kernel terms—then the existence clause of Theorem 1.1 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is almost-global existence, which includes existence and uniqueness of a classical solution, not merely a priori stability of solutions that are assumed to exist. The proof's bootstrap (Lemma 7.13 and the concluding paragraph of Section 7.3) requires a local well-posedness theory for (1.9) in H^{s+1/4}_0 × Hdot^{s-1/4}. This is not proved in the manuscript. Remark 7.11 states that local existence can be derived following [25] 'with minor modifications' and cites [3,53] as alternates. That transfer is load-bearing: [25] treats Euler–Korteweg equations on T^d, not the Kelvin–Helmholtz system with nonlinear singular integral operators H(η), D0(η), K(η) of (1.5); [3,53] treat vortex sheets with surface tension in different geometric settings and without the rotating-frame, zero-mean, and homogeneous-quotient structure used here. Theorem 4.2 paralinearizes (1.9) only for functions that already solve it, so it cannot by itself supply existence. The non-standard regularity gap (η in H^{s+1/4}, ψ in H^{s-1/4}), the zero-average constraint on η, and the possible loss of derivatives in the quasilinear terms B_b and V_b must all be checked in a written symmetrizer or tame-energy argument. Unless that check is supplied, the lifespan estimate is an estimate for a class of solutions whose existence in the stated space has not been established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the 2D Kelvin-Helmholtz system for vortex sheets near a circular interface, in the contour-dynamics formulation (1.9). The main result, Theorem 1.1, asserts that for every Weber number beta = b^2/gamma in [beta1,beta2] subset (0,4(2+sqrt(3))) outside a measure-zero exceptional set, and for every N, small initial data (eta0,psi0) in H^{s+1/4}_0 x \\dot{H}^{s-1/4} admit a unique classical solution that remains O(epsilon)-close to the circular state for |t| <= c epsilon^{-(N+1)}. The proof proceeds through a Hamiltonian formulation (Section 2), a linear stability analysis and a Delort-Szeftel-based non-resonance argument (Section 2.3), a paralinearization of the singular integral operators H, D0, H0 (Section 4), a complex Hamiltonian formulation (Section 5), a paradifferential reduction to constant coefficients (Section 6), and a Hamiltonian Birkhoff normal form with super-action-preserving structure (Section 7). The final energy estimate (Lemma 7.13) yields the lifespan bound.","tokens_in":84592,"tokens_out":22940,"duration_ms":221026,"significance":"If valid, Theorem 1.1 would prove the 1991 numerical conjecture of How-Lowengrub-Shelley and would provide the first almost-global nonlinear stability theorem for the Kelvin-Helmholtz system near a non-flat equilibrium, showing that the combined effect of a background velocity jump and surface tension can suppress nonlinear destabilization. The paper's strengths are substantial: the result is parameter-free in the sense that the Weber number is a physical input rather than a fitted parameter, the exceptional set is proved to be measure zero via an external theorem, and the paralinearization of the Birkhoff-Roth singular integral operators in Section 4 is carried out with explicit proofs. The main caveat is that local well-posedness, which is part of the statement of Theorem 1.1, is not proved in the manuscript.","major_comments":[{"comment":"The existence and uniqueness clause of Theorem 1.1 is load-bearing but is not proved in the manuscript. The bootstrap in Section 7.3, in particular Lemma 7.13, starts from a solution U of (5.9), and the only support for the existence of such a solution is Remark 7.11, which states that local existence can be derived following [25] with minor modifications and cites [3,53] as alternates. These references concern different systems, namely Euler-Korteweg equations on T^d and vortex sheets with surface tension in other geometric formulations, and the manuscript does not verify the transfer to the singular-integral system (1.9) with the nonstandard regularity gap eta in H^{s+1/4}, psi in \\dot{H}^{s-1/4}, the zero-average constraint, and the rotating-frame structure. Since Theorem 1.1 asserts a unique classical solution, this gap must be filled, either by a self-contained local well-posedness theorem or by a precise statement whose hypotheses are checked against the cited results.","section":"Theorem 1.1; Remark 7.11; Section 7.3"}],"minor_comments":[{"comment":"The exceptional set B in Theorem 1.1 is chosen before N, whereas Proposition 2.6 constructs a set that depends on the length bound M through the restriction |alpha+alpha'| <= M. Since the countable union of measure-zero sets is measure-zero, one can take B to be the union over M, but this should be stated explicitly to justify the quantifier order in Theorem 1.1.","section":"Theorem 1.1; Proposition 2.6"},{"comment":"The symbol B^K_{s,R} is defined on page 18 and redefined with a different meaning in Notation 4.1. The authors explicitly warn the reader about this conflict, but a distinct notation would avoid confusion.","section":"Notation 4.1"},{"comment":"Several displayed formulas contain typesetting artifacts in the arXiv text, for example /nabla followed by a marker before sqrt(3) in (1.11) and (2.18), and /radicaltp, /radicalvertex, /bracehtipupleft in Section 4 and Appendix A. The published version should be checked carefully.","section":"Throughout"},{"comment":"The factorization in Eq. (2.36) is hard to read because of missing parentheses; it should be displayed as (n_b - n_a)(n_b - (2n_a-1)/(n_a-2)) up to a nonzero factor, rather than the compressed form currently shown.","section":"Lemma 2.10 and Eq. (2.36)"},{"comment":"The proof of Proposition 7.9 is presented as a sketch that repeatedly refers to [27] for the symplectic correction and the homological equations. Since the Kelvin-Helmholtz system has additional x-dependent order-one and order-one-half terms, a few more details on how Theorem A.16 is applied to the specific map B(U) of Proposition 6.1 would improve verifiability.","section":"Proposition 7.9"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is built largely on the authors' previous framework in [19,27,61], which is natural for this program. However, because Remark 7.11 delegates local well-posedness to a paper co-authored by two of the present authors ([25]) without a written transfer, the editor may want independent scrutiny of the local well-posedness question before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know about arXiv:2504.16861. First, it genuinely targets the How-Lowengrub-Shelley numerical conjecture: for almost every Weber number below the linear threshold, small near-circular vortex sheets survive for time ~ ε^{-(N+1)} for any N. Second, the proof does not contain a local well-posedness theorem for the system (1.9); Remark 7.11 defers to [25] with \"minor modifications\", and that transfer is load-bearing because existence is part of the claimed theorem.\n\nWhat is genuinely new: the paralinearization of the Birkhoff-Roth contour dynamics (Theorem 4.2), the complex Alinhac good unknown, and the first rigorous derivation of the Weber-number threshold and the super-action-preserving normal form in this setting. The non-resonance analysis in Section 2.3 is self-contained up to the Delort-Szeftel theorem, and the measure-zero exceptional set is produced rather than assumed. The Hamiltonian structure is carefully established. This is serious, high-level work.\n\nThe soft spot is real and located exactly where it matters. The bootstrap in Section 7.3 starts from a solution that must exist in H^{s+1/4} × H^{s-1/4}; Theorem 4.2 only paralinearizes solutions that already solve (1.9). The cited [25] treats Euler-Korteweg on T^d, not vortex sheets with the nonlocal singular operators H(η), D0(η), K(η). The alternates [3,53] handle vortex sheets with surface tension in different geometric settings and without the rotating-frame/zero-mean/homogeneous-quotient structure. It is plausible the transfer works, but it is not demonstrated, and the regularity gap (η in H^{s+1/4}, ψ in H^{s-1/4}) is non-standard. This is addressable, not fatal, but it is a genuine gap in the manuscript as written.\n\nSecondary note: large parts of the Birkhoff normal form machinery are quoted from [27], which is acceptable because [27] is published, but a referee will need to trust both papers. The paper is honest about this in its structure.\n\nOverall: the central argument is coherent and the tools are appropriate. The gap is not a fabricated concern—it is explicitly conceded in Remark 7.11, and it sits exactly where the theorem needs it. If the authors supply a written LWP proof, or even a precise theorem statement with a transparent verification of the transfer, this becomes a major paper.\n\nWho it is for: people working on vortex sheets, water waves, Hamiltonian PDEs, and paradifferential methods. It deserves a serious referee—I would send it to review without hesitation, with instructions to focus on the LWP transfer and the non-resonance constants. My own verdict at this stage would be conditional.","headline":"A serious, major-result paper whose central existence clause is not proved in the manuscript: the LWP transfer in Remark 7.11 is load-bearing and needs to be written out.","tokens_in":85172,"tokens_out":2350,"would_cite":true,"duration_ms":23858,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B35","35Q31","35S50","37K55","76B47"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves almost-global existence for nearly circular Kelvin-Helmholtz vortex sheets: for almost every Weber number below $4(2+\\sqrt{3})$, small perturbations persist for time $c\\varepsilon^{-(N+1)}$.","keywords":["Kelvin-Helmholtz instability","vortex sheets","surface tension","Weber number","Hamiltonian Birkhoff normal form","paradifferential calculus","singular integral operators","almost global existence"],"falsifier":"Take a small initial datum of size $\\varepsilon=10^{-3}$ in (1.4), choose $\\beta=5$ below $4(2+\\sqrt3)$ and $N=1$, and integrate numerically up to $t=c\\varepsilon^{-2}$; if the amplitude grows past $C\\varepsilon$ or the active Fourier spectrum spreads beyond the first few modes before that time, the claimed lifespan estimate is false.","tokens_in":84060,"feed_emoji":"🌀","tokens_out":12425,"duration_ms":118801,"temperature":0.7,"pith_summary":"The paper investigates the two-dimensional Kelvin-Helmholtz system for the interface between two irrotational, same-density fluids, written in rotating coordinates as a contour-dynamic equation for $(\\eta,\\psi)$. Its goal is to show that small perturbations of the circular vortex sheet with background velocity jump $b$ and surface tension $\\gamma$ remain close to the circle for very long times. The main theorem states that for almost every ratio $\\beta=b^2/\\gamma$ in any interval inside $(0,4(2+\\sqrt{3}))$, and for any prescribed $N$, initial data of size $\\varepsilon$ in $H^{s+1/4}_0\\times\\dot H^{s-1/4}$ produce a unique classical solution staying within $C\\varepsilon$ of the circle up to time $c\\,\\varepsilon^{-(N+1)}$. A sympathetic reader should care because this confirms a 1991 numerical prediction that motion below a critical Weber number remains predictable over long times, and because it gives the first nonlinear long-time stability theorem for the Kelvin-Helmholtz system. The decisive point is that capillarity alone stabilizes only at linear order, whereas the combination of velocity jump and capillarity tunes the linear frequencies and suppresses resonances.","feed_headline":"Small vortex sheets stay near circular for almost arbitrarily long times","feed_subtitle":"For almost every Weber number below a sharp threshold, small perturbations survive far beyond the local-in-time scale.","key_machinery":"The load-bearing object is the Weber number $\\beta=b^2/\\gamma$, the ratio of the squared background velocity jump to surface tension, which appears in the equilibrium spectrum $\\omega_{\\gamma,b}(\\xi)$ and determines the linear stability threshold $4(2+\\sqrt3)$. The proof chains several mechanisms: a Hamiltonian formulation $H=E_b+\\gamma L+\\Omega M$ with symplectic structure $J\\nabla H$; a new paralinearization of the singular integral operators $H(\\eta)$ and $D_0(\\eta)$ by Taylor-expanding their convolution kernels in the shift variable around $z=0$, converting them into paradifferential symbols plus smoothing remainders; complex coordinates, a good-unknown change, block diagonalization, and reduction to constant coefficients up to smoothing remainders; and a Hamiltonian Birkhoff normal form in which the only surviving resonances are the super-action-preserving monomials, for which the quantities $|u_j|^2+|u_{-j}|^2$ are conserved. A small-divisor non-resonance result, quoted in the paper as Theorem 2.7, controls the quasi-resonances at arbitrary order for $\\beta$ outside a zero-measure set, and the super-action conservation is what makes the normal-form remainders harmless in the Sobolev energy estimate.","core_discovery":"The central claim is an almost-global existence and stability theorem for the system (1.9) near $(\\eta,\\psi)=(0,0)$. Precisely, for any $N\\in\\mathbb N$, any interval $[\\beta_1,\\beta_2]\\subset(0,4(2+\\sqrt3))$, and any $\\beta=b^2/\\gamma$ outside a zero-measure set $B\\subset[\\beta_1,\\beta_2]$, there is a Sobolev regularity $s_0$ such that every initial datum of size $\\varepsilon$ in $H^{s+1/4}_0\\times\\dot H^{s-1/4}$ admits a unique classical solution with $\\sup_t(\\|\\eta(t)\\|_{H^{s+1/4}_0}+\\|\\psi(t)\\|_{\\dot H^{s-1/4}})\\le C\\varepsilon$ for $|t|\\le c\\varepsilon^{-(N+1)}$. The authors interpret this as proving that the stabilizing effect of capillarity is not washed out by the destabilizing velocity jump: the parameter $\\beta$ modulates the linear frequencies $\\omega_{\\gamma,b}(\\xi)=\\sqrt{\\gamma|\\xi|/2}\\sqrt{|\\xi|^2-(\\beta/2)|\\xi|+\\beta-1}$, and excluding a zero-measure set of $\\beta$ removes resonances, so the Birkhoff normal form can be carried to arbitrary order. In this sense the result proves the numerical conjecture of the 1990s and identifies the first parameter regime in which the nonlinear Kelvin-Helmholtz system is almost-globally well-posed.","pith_inferences":["Beyond the paper, the result does not identify an explicit stable value of $\\beta$; if a Diophantine verification could be done on an interval, the 'almost every' conclusion would be upgraded to 'every' on that interval.","Beyond the paper, the super-action conservation predicts a sharp numerical signature: over the almost-global time scale, the Fourier energies of modes $n$ and $-n$ should remain nearly equal, a prediction the paper does not itself test with simulations.","Beyond the paper, the same parameter-modulation strategy may apply to other Hamiltonian vortex-interface models with a tunable physical parameter, but any such transfer would require an independent non-resonance and local-well-posedness check."],"forward_implications":["For any fixed $N$, solutions exist and stay close to the circular state on the time scale $\\varepsilon^{-(N+1)}$; taking $N$ larger makes the lifespan an arbitrarily long polynomial in $\\varepsilon^{-1}$, so the result is genuinely almost global rather than local-in-time.","On these time scales, the Fourier modes exchange energy only through super-action-preserving pairs, so the spectral width of a small perturbation remains localized; this gives a dynamical explanation of the slow spectral spreading observed numerically in the 1990s.","The linear stability threshold $4(2+\\sqrt{3})$ is shown to be relevant nonlinearly: below it and away from the resonant set the system is nonlinearly stable, whereas above it the linear spectrum is not purely imaginary and the normal-form argument does not apply.","The phenomenon requires both a non-zero velocity jump and surface tension: when $\\beta=0$, the mechanism fails, so pure-capillarity vortex sheets do not inherit the long-time stability claimed here.","The exceptional set $B$ has zero measure, so the theorem covers almost every Weber number in any prescribed sub-threshold interval."],"supporting_citations":[{"why":"provides the 1991 numerical conjecture of long-time predictability below a critical Weber number that the theorem proves.","marker":"[50]"},{"why":"supplies the Hamiltonian Birkhoff normal form method for quasi-linear systems on which the proof's normal-form step is built.","marker":"[27]"},{"why":"supplies the paralinearization technique for nonlinear singular integral operators that Section 4 generalizes.","marker":"[19]"},{"why":"provides the paradifferential reduction-to-constant-coefficients toolbox used to remove spatial dependence in highest-order terms.","marker":"[20]"},{"why":"introduces the contour-dynamic equation (1.4) in rotating coordinates and the steady circular states used as reference solutions.","marker":"[61]"},{"why":"is the source invoked in Remark 7.11 for the local well-posedness theory from which the existence clause of Theorem 1.1 is derived.","marker":"[25]"},{"why":"establishes local well-posedness of vortex sheets with surface tension, cited as the classical short-time existence framework for the system.","marker":"[3]"},{"why":"provides the two-fluid interface stability criterion and an alternate local existence setting mentioned in Remark 7.11.","marker":"[53]"},{"why":"furnishes the small-divisor non-resonance theorem that controls quasi-resonances and defines the zero-measure exceptional set B.","marker":"[38]"}],"fun_headline_variants":["Capillarity and velocity jump tame vortex sheets for almost all time","Almost global stability for vortex sheets near circular equilibrium","Vortex sheets survive near circular for almost arbitrary long times","Weber threshold yields almost eternal vortex sheet stability","Capillary-stabilized vortex sheets defy Kelvin-Helmholtz almost globally"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that local well-posedness for the nonlinear system in $H^{s+1/4}_0\\times\\dot H^{s-1/4}$ can be obtained by following the argument of the cited local-existence paper [25] with minor modifications; the main theorem's existence clause depends on that transfer, yet the paper only asserts it in Remark 7.11 without writing the proof.","fun_headline_variants_meta":{"raw":{"variants":["Capillarity and velocity jump tame vortex sheets for almost all time","Almost global stability for vortex sheets near circular equilibrium","Vortex sheets survive near circular for almost arbitrary long times","Weber threshold yields almost eternal vortex sheet stability","Capillary-stabilized vortex sheets defy Kelvin-Helmholtz almost globally"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00077,"raw_usage":{"total_tokens":3489,"prompt_tokens":1099,"completion_tokens":2390,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":715,"completion_tokens_details":{"reasoning_tokens":2308}},"tokens_in":715,"tokens_out":2390,"duration_ms":18504,"temperature":1.0,"reasoning_tokens":2308,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:53:29.809448+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small initial datum of size $\\varepsilon=10^{-3}$ in (1.4), choose $\\beta=5$ below $4(2+\\sqrt3)$ and $N=1$, and integrate numerically up to $t=c\\varepsilon^{-2}$; if the amplitude grows past $C\\varepsilon$ or the active Fourier spectrum spreads beyond the first few modes before that time, the claimed lifespan estimate is false.","supporting_citations":[{"cited_title":"How, John S","cited_arxiv_id":null,"evidence_quote":"provides the 1991 numerical conjecture of long-time predictability below a critical Weber number that the theorem proves."},{"cited_title":"Hamiltonian Birkhoff normal form for gravity-capillary water waves with constant vorticity: almost global existence","cited_arxiv_id":null,"evidence_quote":"supplies the Hamiltonian Birkhoff normal form method for quasi-linear systems on which the proof's normal-form step is built."},{"cited_title":"Paralinearization and extended lifespan for solutions of the α-SQG sharp front equation","cited_arxiv_id":null,"evidence_quote":"supplies the paralinearization technique for nonlinear singular integral operators that Section 4 generalizes."},{"cited_title":"Almost global solutions of capillary-gravity water waves e quations on the circle , volume 24 of Lecture Notes of the Unione Matematica Italiana","cited_arxiv_id":null,"evidence_quote":"provides the paradifferential reduction-to-constant-coefficients toolbox used to remove spatial dependence in highest-order terms."},{"cited_title":"Local well posedness of the Euler-Korteweg equations on Td","cited_arxiv_id":null,"evidence_quote":"is the source invoked in Remark 7.11 for the local well-posedness theory from which the existence clause of Theorem 1.1 is derived."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes local well-posedness of vortex sheets with surface tension, cited as the classical short-time existence framework for the system."},{"cited_title":"A stability criterion for two-fluid interfaces an d applications","cited_arxiv_id":null,"evidence_quote":"provides the two-fluid interface stability criterion and an alternate local existence setting mentioned in Remark 7.11."},{"cited_title":"Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres","cited_arxiv_id":null,"evidence_quote":"furnishes the small-divisor non-resonance theorem that controls quasi-resonances and defines the zero-measure exceptional set B."}],"review_version":1}