{"id":"46e1daa2-696f-4cc8-8856-98450f89839c","arxiv_id":"2608.13134","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every m at least 12, there exist m-fold symmetric uniformly rotating vortex patches whose boundaries have exactly m right-angle corners, proved by a new fixed-point construction.","lead":"This paper proves that uniformly rotating vortex patches with sharp 90-degree corners exist for all m-fold symmetries with m at least 12 in the 2-D incompressible Euler equations. It is the first rigorous existence proof for singular rotating patches, settling a long-open conjecture in the large-symmetry regime.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Omitted derivative estimate (5.29) in Lemma 5.5 is the load-bearing gap: Proposition 5.4 needs it to prove the C^1 endpoint closure f'(0±)=∓1; without a complete derivation the boundary-regularity claim in Theorem 1.1(iii)/(v) is unproved.","rationale":"The reader's weakest assumption identifies precisely the same load-bearing spot: the unproved estimate (5.29) in Lemma 5.5, which Proposition 5.4 uses to control the denominator in (5.37). I agree with that identification. The concern is genuine because the paper itself marks the calculation as omitted, and because the asserted C^1 closure of the patch boundary at the corner—not merely the existence of a 90-degree angle through one-sided slopes—depends on it. I would not escalate to REJECT: the surrounding argument is detailed, the fixed-point construction and the first-order expansions (5.27)–(5.28) are written out, and the missing estimate is described as following the same method already used, so a repair is credible. The paper also contains substantial independent support: the result is consistent with the necessary condition in [82], with the formal analysis in [54], and with the quantitative estimates in [72], and the explicit threshold m0 ≤ 12 is backed by a lengthy but explicit Appendix A computation. Therefore the reader's CONDITIONAL verdict is appropriate, and my stress-test does not change it.","tokens_in":102607,"tokens_out":10585,"duration_ms":383667,"concrete_test":"Write out the missing derivation of (5.29): apply Lemma 2.2, set (x_1,x_2) = (π − r cos α, r sin α), and split the integral by the same S_{3r} sector decomposition used for (5.27). Isolate the contribution of ∂_{x2} of the harmonic remainder K and verify it is O(r); then expand the remaining term and check that the coefficient of r ln r is exactly (sin α / π) A(r; g). If this coefficient matches, re-run the limit in Proposition 5.4; if the O(r) bound fails or the coefficient differs, Theorem 1.1(iii)/(v) must be weakened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 5.5 states three local expansions near the corner (π,0). The first two, (5.27) and (5.28), are proved in detail. The third, (5.29), controlling ∂_{x2}Ψ − ∂_{x2}Ψ(π,0), is introduced with the sentence: “One can show (5.29) in a similar way by studying d_{x2}W... We omit the tedious details for brevity.” This estimate is not decorative. Proposition 5.4 derives the derivative identity g'(π−) = −1 from (5.37), whose denominator is ∂_{x2}Ψ(x,g(x)) − ∂_{x2}Ψ(π,0)e^{−2μg(x)}. Controlling that denominator at the sharp r|ln r| scale is exactly what (5.29) supplies; the numerator uses (5.28). Without (5.29), the limit of g'(x) as x→π− is not justified, so Theorem 1.1(iii) (f ∈ C^1([0, 2π/m])) and (v) (f'(0±) = ∓1) lose their proof. Note that the 90-degree angle itself, obtained from the difference-quotient slopes g'_−(π) = −1 and g'_+(−π) = 1, does not require (5.29); the more exposed part is the C^1-closure claim. The omitted-calculation pattern also recurs in the analyticity bootstrap via [52, Theorem 5.2] and [57, Theorem 3.1'] and in Proposition 6.1(i)–(ii). The missing proof of (5.29) is very plausible and likely repairable, but it is not present in the manuscript.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for every integer m ≥ 12, an m-fold symmetric uniformly rotating vortex patch of the 2-D incompressible Euler equation whose boundary has exactly m corners, each of angle 90°. The construction passes through a conformal change of variables to a strip, reformulates the free-boundary condition as the fixed-point problem R(g) = g for an implicit monotone operator R, and proves existence by combining a Schauder fixed-point argument with explicit upper/lower barrier functions. The authors then establish continuity, interior C^{1,α} and analytic regularity, and endpoint asymptotics that yield the corner slopes f'(0±) = ∓1. They also characterize the modified stream function in the co-rotating frame, the outside streamline connecting neighboring corners, and the sign of the radial and angular velocity. The numerical threshold m0 ≤ 12 is obtained from explicit rational estimates in Appendix A.","tokens_in":102844,"tokens_out":7590,"duration_ms":73517,"significance":"If the missing estimates are supplied, this is a substantial advance: it provides the first rigorous construction of rotating Euler vortex patches with singular boundary, confirming a conjecture in Hassainia–Masmoudi–Wheeler and matching the necessary right-angle condition of Wang–Zhang–Zhou. The fixed-point construction is genuinely self-contained: the constants M = 4, Λ*, λ* are derived from the problem data rather than fitted to the target solution, and the explicit-constant verification in Appendix A is unusually careful and reproducible. The theorem also gives concrete quantitative information: corner locations, boundary asymptotics, and two-sided bounds on angular velocity. These falsifiable predictions are a genuine strength. The main obstacles to accepting the paper as written are the omitted proof of a load-bearing local expansion, the corresponding omissions in the streamline theorem, and the sketchy analyticity bootstrap.","major_comments":[{"comment":"The estimate (5.29) is stated but not proved; the text says \"One can show (5.29) in a similar way by studying d_x2 W... We omit the tedious details for brevity as the argument is largely the same.\" This estimate is load-bearing: Proposition 5.4 derives the formula (5.37) for g'(x), and the denominator there is ∂_{x2}Ψ(x,g(x)) − ∂_{x2}Ψ(π,0)e^{-2µg(x)}. The sharp r|ln r| control supplied by (5.29) is exactly what allows the limit g'(π−) = −1 to be extracted. Without (5.29), the claims f ∈ C^1([0,2π/m]) in Theorem 1.1(iii) and the endpoint derivative identities in Theorem 1.1(v) are not established. I want to be precise: the 90-degree angle itself, as a difference-quotient slope, follows from the r|ln r| expansion already proved in (5.27) and (5.28); the missing point concerns the C^1-closure of the boundary curve. The authors must supply a complete proof of (5.29), even if it is placed in an appendix.","section":"§5.3, Lemma 5.5, Eq. (5.29)"},{"comment":"The proof of Lemma 5.6 is omitted (\"The proof is completely parallel to that of Lemma 5.5... We omit the details\"), and Proposition 6.1(ii) is justified by saying the endpoint slopes can be shown \"following exactly the same idea of proving Proposition 5.4 by Lemma 5.5.\" These statements are not cosmetic: they are the basis for Theorem 1.2(i)–(iv), including the C^1 regularity and endpoint slopes of the outer streamline f̂, and they feed into the sign characterizations in Theorem 1.3. Moreover, Lemma 5.6 is a genuinely separate expansion for Ψ2, so the omission compounds the missing (5.29). The proof should be included, or Theorem 1.2 should be reformulated as conditional on a stated lemma with full proof.","section":"§6, Proposition 6.1 and Lemma 5.6"},{"comment":"The analyticity claim in Theorem 1.1(iii) is justified by invoking [57, Theorem 3.1'] and [52, Theorem 5.2] with only a sketch. The hypotheses of [57, Theorem 3.1'] are not stated, and the verification of condition (c) (uniform C^2 up to the boundary on each side) is only sketched, with the final estimates referred to [52]. Since the free boundary here has C^{1,α} regularity but is not uniformly smooth, the reader cannot check from the manuscript that the quoted theorem applies verbatim. Either state the external theorem with its exact hypotheses and verify them in this setting, or give a self-contained bootstrap. This matters because analyticity is part of the stated main theorem, even though it is not needed for the existence of the corner.","section":"Proof of Theorem 1.1, analyticity of f in (0, 2π/m)"}],"minor_comments":[{"comment":"The intervals in Table 1 are described as \"numerically found,\" and Remark 5.8 uses them to discuss why µ = 1/12 is the only admissible case among the values listed. The final theorem does not rely on these numerics, but the paper should state explicitly in Remark 5.8 that Table 1 is heuristic and not used in the proof of Corollary 3.1.","section":"§3.1, Table 1 and Remark 5.8"},{"comment":"The notation g'_−(π) is used both for the difference quotient limit lim_{x→π−} g(x)/(x−π) and for the derivative limit lim_{x→π−} g'(x). The two are defined in the proof, but the parallel notation in the abstract statements of Theorem 1.1(v) should be made uniform so that the reader does not confuse the angle slope with the C^1 derivative endpoint value.","section":"§5.3, Proposition 5.4"},{"comment":"The numerical section is presented as an illustration and is not used in the proof, but a short sentence in Section 1 or Appendix C explicitly stating that Appendix C plays no role in the rigorous theorems would help prevent the numerical intervals in Table 1 from being read as part of the proof.","section":"Appendix C"},{"comment":"Remark 6.2 says the µ = 0 case of Proposition 6.1 can be handled in a completely parallel way and omits the proof. Since the paper already treats µ > 0 in the main text and the formal limit µ = 0 is used only as a heuristic, this is acceptable, but the remark should make clear that no theorem for µ = 0 is being claimed.","section":"§6, Remark 6.2"}],"recommendation":"major_revision","confidential_remarks":"This is a serious paper with an unusually detailed proof structure, and the omitted estimates are very plausibly repairable. My recommendation is major revision rather than reject because the missing proof of (5.29), the parallel omissions in Proposition 6.1/Lemma 5.6, and the gap between the cited analyticity theorem and its verification are all fixable within the manuscript's scope. I would advise the editor that the paper should not be accepted until those proofs are supplied."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is the first rigorous construction of uniformly rotating vortex patches with 90-degree corners, for all m ≥ 12. The proof is detailed and mostly self-contained, but one load-bearing local expansion is stated without proof. Read it with that caveat in mind.\n\nThe existence result is genuinely new. Prior knowledge was numerical observation [83], formal asymptotics [54], necessary conditions [82], and the open conjecture [46]. The paper gives a direct fixed-point construction: monotonicity of the renormalized stream function, upper/lower barriers, Schauder fixed point, then regularity. It avoids bifurcation from the disk entirely. The explicit threshold m0 ≤ 12 is backed by rigorous rational bounds in Appendix A, which is credible even though I haven't machine-checked it. The constants are proved admissible, not fitted to a solution, so circularity is not a concern.\n\nThe soft spot is real: Lemma 5.5 states the local expansion (5.29) with \"we omit the tedious details for brevity.\" Proposition 5.4 uses exactly that estimate to control the denominator of (5.37) and justify the C^1 closure f'(0±)=∓1 in Theorem 1.1(iii)/(v). Without a complete proof of (5.29), that part of the main theorem is unproved as written. The 90-degree angle itself—the one-sided slopes g'_-(π)=-1—does not require (5.29), but the C^1-closure does. The estimate is plausible and likely repairable, but it is not present. The same pattern shows up in the analyticity bootstrap (deferred to [52, Theorem 5.2] and [57, Theorem 3.1']) and in Proposition 6.1(i)-(ii), where the corner behavior of the outer streamline is asserted with \"we omit the details.\" Those are less concerning because they reference standard arguments, but (5.29) is load-bearing.\n\nIf the omitted estimates close as claimed, this is a substantial advance: first singular V-states, consistent with [82], relevant to [46]. The paper is clearly written, the main structural steps are proved, and the failures are of completeness, not of coherence. The numerics in Remarks 3.1 and 5.8 are honestly labeled as heuristic.\n\nAnyone working on V-states, free boundary problems for Euler, or singular vortex patch dynamics will get value from this. It deserves a serious referee. I'd send it to review, with the expectation that the authors supply a full proof of (5.29) and the other sketch-and-reference steps.","headline":"First rigorous construction of rotating vortex patches with 90-degree corners, with a repairable but load-bearing omitted estimate in the corner-regularity proof.","tokens_in":103534,"tokens_out":2632,"would_cite":true,"duration_ms":25071,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q31","35R35","76B47"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that for every integer m≥12 there exists an m-fold symmetric uniformly rotating vortex patch of the 2-D Euler equation whose boundary has exactly m 90-degree corners, is C^1 on each closed arc and analytic in the interior.","keywords":["vortex patch","V-state","Euler equations","90-degree corner","m-fold symmetry","fixed-point method","free boundary","co-rotating frame"],"falsifier":"Solve the fixed-point equation of Section 2 numerically for m=12 to high precision and measure lim_{r→0+}g(π−r)/r; if it is not 1, the 90-degree angles fail. Alternatively, prove or disprove estimate (5.29) directly from the integral formula for ∂x2Ψ; a different leading coefficient would invalidate the slope extraction.","tokens_in":102241,"feed_emoji":"🌀","tokens_out":9294,"duration_ms":86967,"temperature":0.7,"pith_summary":"The paper proves that planar Euler flow admits uniformly rotating vortex patches whose boundaries genuinely contain 90-degree corners—a configuration previously observed numerically and predicted by formal asymptotics but never constructed rigorously. The result covers every m-fold symmetric patch with m≥12, giving a boundary that is $C^{1}$ on each arc, analytic between corners, and exactly m corners at prescribed points. It also gives quantitative bounds on the angular velocity and a complete description of the stationary flow in the co-rotating frame, including a second streamline outside the unit disk that meets the patch boundary at each corner. The proof works by recasting the free-boundary equation as a fixed point of a monotone implicit map in a logarithmic strip, then extracting the corner from a local expansion of the stream function.","feed_headline":"For m≥12, rotating vortex patches with 90-degree corners exist","feed_subtitle":"A fixed-point proof gives singular vortex patches with right-angle corners that were only conjectured before.","key_machinery":"The central object is a fixed point g of an implicit map R defined on profiles in a logarithmic strip. After the conformal change ζ(r cosθ, r sinθ)=(mθ−π, −m ln r), the patch boundary becomes the graph of g, and the stream function is encoded by an integral Ψ over the region under this graph. Setting F=Ψ/x2, the map R is defined by the equation F(x1,R(g)(x1);g)=∂x2Ψ(π,0;g)·(1−$e^{{−2µg(x1)}}$)/(2µg(x1)), with µ=1/m. The key properties are strict monotonicity of F in both variables on [0,π]×(0,∞), which makes R single-valued, and explicit upper and lower barrier functions that confine the fixed point. A Schauder argument gives the fixed point, a non-degeneracy estimate for ∂x2F† yields continuity and $C^{{1,α}}$ regularity, and a local expansion of Ψ near (π,0) fixes the one-sided slopes of g, which in physical coordinates become right angles.","core_discovery":"The central discovery is Theorem 1.1: there exists a universal m0≤12 such that for every integer m≥12 there is an f∈M0∩C(T) for which D0(f) is a uniformly rotating vortex patch of the 2-D Euler equation with m-fold symmetry. The boundary is $C^{1}$ on each closed arc and analytic in the interior, and it has exactly m corner points at the directions (cos(2kπ/m), sin(2kπ/m)) where the two boundary arcs meet at a 90-degree angle. The angular velocity a satisfies −4/m ≤ a−1/2 ≤ −Cλ/m. The companion results characterize the co-rotating flow: the level set of the modified stream function through each corner consists of the patch boundary inside the unit disk together with a unique outer streamline outside the disk, meeting at four right angles at each corner, and the set of stagnation points is finite and confined to symmetry lines.","pith_inferences":["The numerical appendix finds fixed points for all µ∈[0,1/2], i.e. m≥2, so the m≥12 threshold is likely an artifact of the constants rather than a true border; a refined bookkeeping may push the proof below m=12.","If the numerical uniqueness is genuine, the constructed family is a strong candidate for the limiting configurations predicted along global bifurcation branches.","These patches are rotating analogues of the fragile 90-degree corners studied in ill-posedness results; their existence suggests that rotation can stabilize a singular boundary that would not persist in a general evolution, and it would be worth testing whether nearby non-symmetric perturbations preserve the corners.","The cat's-eye structure of Theorem 1.2 leaves exactly one unresolved stagnation point in each eye; a finer geometric estimate on the two boundary curves might settle the cat's-eye conjecture."],"forward_implications":["For every m≥12 there is an honest V-state with a nonempty singular set, realizing the necessary conditions previously derived for rotating vortex patches and confirming that the free-boundary analysis is not vacuous.","The patch boundary is C^1 along each closed arc, analytic between corners, and has exactly m corners of interior angle 90°, giving the first concrete shapes in which 90-degree corners persist in a Euler flow.","The angular velocity satisfies −4/m ≤ a−1/2 ≤ −Cλ/m, so for large m the patch rotates just below the half-unit angular speed of the disk family, with the correction of order 1/m.","In the co-rotating frame the level set through each corner consists of two C^1 curves, one inside and one outside the unit disk, meeting at four right angles; the radial velocity has a fully determined sign pattern on the whole plane away from the origin.","The set of stagnation points is finite, and for all sufficiently large m the azimuthal velocity is positive everywhere in the patch except at the origin.","The construction settles a previously open conjecture that m-fold symmetric V-states with 90-degree corners exist; the built family is the first rigorous example of a rotating vortex patch with singular boundary.","Because the boundary is analytic between corners but only C^1 at the corners themselves, the example pins down exactly how singular the co-rotating streamlines can be while still solving the Euler equation.","The presented fixed-point strategy is parameter-free and supplies quantitative estimates on the patch profile and angular velocity, so the same method can be adapted to search for other singular free boundaries in the 2-D Euler equation."],"supporting_citations":[{"why":"Supplies the fixed-point construction and the analytic bootstrap used to find the patch profile and to upgrade interior regularity.","marker":"[52]"},{"why":"Established the necessary-condition result that Lipschitz V-state boundaries meet at right angles at finitely many singular points; the constructed patches realize that condition.","marker":"[82]"},{"why":"Posed the conjecture that m-fold V-states with 90-degree corners arise as weak limits; this paper constructs them directly.","marker":"[46]"},{"why":"Gave the formal asymptotic analysis identifying 90 degrees as the only possible corner angle for limiting patches.","marker":"[54]"},{"why":"Supplies the free-boundary analyticity theorem used to conclude that the patch boundary is analytic between corners.","marker":"[57]"},{"why":"Numerically observed limiting V-states with corners, the phenomenon this paper makes rigorous.","marker":"[83]"}],"fun_headline_variants":["Rotating vortex patches with right-angle corners exist for all m≥12","90-degree corner vortex patches proven for m-fold symmetry, m≥12","Uniformly rotating patches with right-angle corners: existence for m≥12","Vortex patches with 90-degree corners rotate uniformly for m≥12","Existence of 90-degree corner vortex patches for m≥12"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The 90-degree corner conclusion depends on the omitted derivative estimate (5.29), which fixes the limiting boundary slope at each corner; if that calculation does not close, the angle statement loses its proof.","fun_headline_variants_meta":{"raw":{"variants":["Rotating vortex patches with right-angle corners exist for all m≥12","90-degree corner vortex patches proven for m-fold symmetry, m≥12","Uniformly rotating patches with right-angle corners: existence for m≥12","Vortex patches with 90-degree corners rotate uniformly for m≥12","Existence of 90-degree corner vortex patches for m≥12"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000753,"raw_usage":{"total_tokens":3269,"prompt_tokens":780,"completion_tokens":2489,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":396,"completion_tokens_details":{"reasoning_tokens":2394}},"tokens_in":396,"tokens_out":2489,"duration_ms":15042,"temperature":1.0,"reasoning_tokens":2394,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:01:54.798033+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the fixed-point equation of Section 2 numerically for m=12 to high precision and measure lim_{r→0+}g(π−r)/r; if it is not 1, the 90-degree angles fail. Alternatively, prove or disprove estimate (5.29) directly from the integral formula for ∂x2Ψ; a different leading coefficient would invalidate the slope extraction.","supporting_citations":[{"cited_title":"Steady contiguous vortex-patch dipole solutions of the 2D incompressible Euler equation.Archive for Rational Mechanics and Analysis, 249(46), 2025","cited_arxiv_id":null,"evidence_quote":"Supplies the fixed-point construction and the analytic bootstrap used to find the patch profile and to upgrade interior regularity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Established the necessary-condition result that Lipschitz V-state boundaries meet at right angles at finitely many singular points; the constructed patches realize that condition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Posed the conjecture that m-fold V-states with 90-degree corners arise as weak limits; this paper constructs them directly."},{"cited_title":"Overman II","cited_arxiv_id":null,"evidence_quote":"Gave the formal asymptotic analysis identifying 90 degrees as the only possible corner angle for limiting patches."},{"cited_title":"Regularity in elliptic free boundary problems I.Journal d’Analyse Math´ ematique, 34(1):86–119, 1978","cited_arxiv_id":null,"evidence_quote":"Supplies the free-boundary analyticity theorem used to conclude that the patch boundary is analytic between corners."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Numerically observed limiting V-states with corners, the phenomenon this paper makes rigorous."}],"review_version":1}