{"id":"a7b4c6ba-d834-48dd-b01c-80ebd73c0965","arxiv_id":"2411.14859","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the two-dimensional zero-surface-tension contact Muskat problem with acute contact angles, local classical solvability in weighted Hölder spaces and the waiting time phenomenon are established under stated assumptions.","lead":"The paper proves that the two-phase Muskat problem, where an interface between fluids meets fixed walls at acute angles, has a unique classical solution for a short time. It also gives conditions under which the two corner contact points remain fixed, a behavior called the waiting time effect.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.1's a priori estimate appears to misapply Lemma 5.1(iii): the flux datum φ2 in (5.17) is nonzero, while the lemma requires φ2 ≡ 0; without this bound the linear theory and contraction collapse.","rationale":"The paper's central claim is local well-posedness of the two-phase contact Muskat problem. The proof reduces the nonlinear problem to a linear transmission problem with dynamic boundary condition (5.13), whose solvability and a priori estimates are the heart of Theorem 3.1. I examined the proof of Theorem 5.1 and found a concrete point where a lemma is invoked outside its hypotheses. In bounding the time-Hölder seminorms of w1, w2, the text says to apply Lemma 5.1(iii) 'where we set φ2 = A0σ'. But in Lemma 5.1, φ2 is the flux datum in the transmission condition k1 ∂W1/∂n = k2 ∂W2/∂n + φ2, whereas A0σ is the trace jump φ1 = W1 − W2. For problem (5.17), the trace jump is indeed A0σ, but the flux datum is φ2 = k^ε A3(∂w1/∂ω − ∂w2/∂ω), which is not zero. Lemma 5.1(iii) explicitly requires φ2 ≡ 0. Thus the bound (5.6) cannot be applied directly; the extra φ2 term must be controlled and shown to be absorbable with a small T factor. The paper does not provide this. This is not a dispute about the truth of the theorem; it is a missing argument in the presented proof of the key linear estimate. The reader's concern about Lemma 4.4 is also real and points in the same direction: several central technical estimates are either omitted or deferred to [35]. My concern is more localized and more immediately checkable. A fix would likely require an additional a priori estimate for the lower-order tangential term in (5.17). The recommended verdict remains CONDITIONAL, because the gap is potentially repairable and the overall strategy is plausible, but the paper in its current form does not fully establish Theorem 5.1 as written.","tokens_in":45089,"tokens_out":12700,"duration_ms":117808,"concrete_test":"Re-derive the estimate for ⟨w1⟩_t and ⟨w2⟩_t in the proof of Theorem 5.1 using the actual data from (5.17): φ1 = A0σ and φ2 = k^ε A3(∂w1/∂ω − ∂w2/∂ω). If the resulting right-hand side contains a term involving ‖∂φ2/∂t‖ or ‖φ2‖ that is not multiplied by a small T^{1−β} factor and cannot be absorbed into the left side of (5.15), then Theorem 5.1 has a genuine gap. A minimal analytic check is to test whether Lemma 5.1(iii) extends to φ2 ≠ 0 with an additional lower-order term; if it does not, the current proof is incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of the central linear estimate in Theorem 5.1 (Section 5.3, paragraph (II)) completes the a priori bound for the continuation family (5.17) by invoking Lemma 5.1(iii). The paper states: 'we utilize (iii) in Lemma 5.1 to (5.17) ... where we set φ2 = A0σ and Wi = wi'. But Lemma 5.1(iii) requires φ0,i, φ2, φ3, φ4 ≡ 0 and bounds ∂Wi/∂t in terms of ∂φ1/∂t. In (5.17) the correct data are φ1 = A0σ (the trace jump w1 − w2) and φ2 = k^ε A3(∂w1/∂ω − ∂w2/∂ω) (from the fourth line of (5.17)); φ2 is generally nonzero. Identifying φ2 with A0σ conflates the trace datum with the flux datum. Consequently the bound (5.6) is invoked outside its hypotheses. If the extra φ2 term cannot be controlled with a small T^{1−β} factor and absorbed into the left side of (5.15), then the a priori estimate for the inverse of the linear operator is not established, and the contraction argument in Section 5.4 loses its foundation. This is a more immediate gap than the deferred estimates in Lemma 4.4, although both concern the same load-bearing linear theory.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the two-dimensional contact Muskat problem with zero surface tension, where the free boundary forms acute corners with the fixed boundary. The main result (Theorem 3.1) asserts local unique classical solvability of the transformed problem (3.9) in weighted H\\\"older spaces under assumptions (h1)-(h7), with the free boundary defined via a Hanzawa-type transformation; Corollary 3.1 states a waiting-time phenomenon. Section 6 extends the result to arbitrary irrational angles in (0,π/4). The proof uses a linearization, a model problem with dynamic boundary conditions in corners, and a contraction argument.","tokens_in":45347,"tokens_out":7816,"duration_ms":69640,"significance":"If the proof is completed, this would be the first local well-posedness result for the two-phase contact Muskat problem with zero surface tension and acute corners, and the waiting-time corollary would address an interesting qualitative question. The paper provides a clear overall strategy and makes the structural assumptions explicit. However, the verification is heavily dependent on prior work of the author ([34]-[38]) and several key estimates are stated without proof, which limits the current reliability.","major_comments":[{"comment":"In bounding the time-Hölder seminorms of w1,w2, the proof invokes Lemma 5.1(iii) for the continuation family (5.17) by setting 'φ2 = A0σ'. This is incorrect: A0σ is the trace datum φ1 (the jump w1−w2), while the flux datum in (5.17) is φ2 = k^ε A3(∂w1/∂ω−∂w2/∂ω), which is generally nonzero and depends on the unknown solution. Lemma 5.1(iii) requires φ2≡0 and yields a bound in terms of ∂φ1/∂t only. Consequently, the bound (5.6) is used outside its hypotheses, and the a priori estimate (5.15) for the inverse operator is not established. Since the contraction argument in Section 5.4 relies on Theorem 5.1, this is a load-bearing gap. Please provide a correct estimate that either absorbs the A3 term into the left side with a small T^{1−β} factor or treats it by a different method.","section":"Section 5.3, Theorem 5.1, paragraph (II)"},{"comment":"Theorem 4.3 concerns the q2=1 case of the corner model, which corresponds to the cases Q0=1 or Q1=1 allowed in (h7). The proof is not provided; the text states 'The proof of Theorem 4.3 is similar to arguments leading to Theorems 4.1 and 4.2. Thus, we verify (here) only the first two theorems.' Because the main theorem explicitly includes the q2=1 cases, this is not a purely cosmetic omission. Either give the proof or restrict the main theorem to cases where the corner model is fully proved.","section":"Section 4, Theorem 4.3"},{"comment":"The a priori estimates for the model problem are central to Theorem 4.1, yet the proof is deferred: after listing asymptotic expansions, the text says 'recasting step by step the arguments of [35, Sections 3-6]' and 'we omit them here.' The problem in [35] is not identical to (4.10) (different coefficients a1,a2,a3,k and different weight), so an explicit statement of how the cited arguments apply, or a full proof, is necessary.","section":"Section 4.3, Lemma 4.4"},{"comment":"The contraction estimates are only cited to [35, Section 5]. Given that the nonlinear terms in Lemma 5.3 depend on the specific geometry and weights of this problem, a brief proof or at least a delineation of the main estimates would be needed to verify the fixed-point argument.","section":"Section 5.4, Lemma 5.4"}],"minor_comments":[{"comment":"The flux condition in (5.17) contains the term k^ε A3(x)(∂w1/∂ω−∂w2/∂ω), but Lemma 5.1, to which the proof refers, has no such term. Please clarify how Lemma 5.1 is adapted to this coefficient or state an analogue for k^ε.","section":"Equation (5.17)"},{"comment":"In the proof of (iii), the text mentions 'the new right-hand side ∂φ2/∂t' although the hypothesis sets φ2≡0; this appears to be a typo for ∂φ1/∂t and should be corrected.","section":"Lemma 5.1(iii)"},{"comment":"The estimates (6.2) are quoted from Theorem 4.2, but Theorem 4.2 itself relies on the unproved Lemma 4.4. Please ensure that the limit argument in Section 6 only uses results that are fully established.","section":"Section 6, around (6.2)"},{"comment":"There are several typographical and formatting issues, such as the title spacing in 'SOL V ABILITY' and the inconsistent spelling of 'Aequationes Mathematicae' in reference [37].","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The manuscript depends on a long series of the author's own prior papers, and a significant portion of the technical proof is delegated to those references. The specific misapplication of Lemma 5.1(iii) in Theorem 5.1 is a serious correctness issue, though I believe it may be repairable. The omission of the proof of Theorem 4.3 is also problematic given that (h7) includes q2=1 cases. I recommend major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a serious paper on a genuinely new problem—two-phase contact Muskat with acute corners and zero surface tension—and the strategy is coherent. But the written proof has a real hole in the linear theory, and several heavy estimates are parked in references or left out. It deserves a referee, not acceptance as is.\n\nWhat the paper does well: the problem is new; previous work covered one-phase contact Hele-Shaw and contactless Muskat, so the first two-phase contact result would be a meaningful step. The weighted-Hölder machinery and the corner model are used carefully, and the waiting-time corollary is a nice touch. The paper is honest about what it leans on.\n\nThe soft spot: the stress-test note is right. In Theorem 5.1, part (II), the paper invokes Lemma 5.1(iii) on the system (5.17) and says it sets φ2 = A0σ. But φ2 in Lemma 5.1 is the flux datum; in (5.17) the flux condition is ∂w1/∂n − k^ε ∂w2/∂n − k^ε A3(...)=0, so the appropriate φ2 is k^ε A3(...), which is generally nonzero. Lemma 5.1(iii) requires φ2 ≡ 0. The trace datum A0σ is φ1, not φ2. So the bound (5.6) is used outside its hypotheses, and the extra flux term is not controlled. Unless that term can be absorbed with a T^{1−β} factor and included in (5.15), the a priori estimate for L^{-1} is not established, and the contraction argument in 5.4 loses its base. This is a more immediate gap than the deferred estimates.\n\nThe other soft spots are minor-to-moderate in comparison: Theorem 4.3 (q2=1) is stated without proof; Lemma 4.4's estimates are explicitly omitted, deferring to [35]; and Lemma 5.4's contraction estimates are cited rather than proved. For a paper this technical, that is a lot of weight on prior work. The self-citations are transparent but dense.\n\nMy read: the central theorem may be true, and the strategy is plausible. But the written proof does not close. This is exactly the kind of paper a serious referee should see—not to kill it, but to ask for the gap to be fixed and the omitted estimates supplied.\n\nRecommendation: send it to referees, with a request to check the application of Lemma 5.1(iii) and the q2=1 corner model.","headline":"First two-phase contact Muskat result, but the linear a priori estimate in Theorem 5.1 misapplies Lemma 5.1(iii), and several key estimates are deferred; deserves a referee, not acceptance as is.","tokens_in":45913,"tokens_out":5165,"would_cite":true,"duration_ms":48592,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R35","35J25","35B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the two-phase contact Muskat problem with zero surface tension has a unique classical solution for a short time when the interface meets the fixed boundary at acute corners, and that the corner points stay fixed…","keywords":["contact Muskat problem","zero surface tension","free boundary","weighted Holder spaces","nonsmooth domains","waiting time","local well-posedness","dynamic boundary condition"],"falsifier":"Compute the solution of the linear corner problem (4.9) for a fixed rational angle, say delta = pi/6, using the explicit integral representation (4.12), and numerically evaluate the weighted Holder norm of (u1, u2) for a compactly supported smooth f1; if the norm grows faster than the bound in Lemma 4.4 as the time horizon T is increased, or if any nonzero f1 with zero initial data gives a non-unique solution, the central claim collapses.","tokens_in":44798,"feed_emoji":"🌊","tokens_out":7493,"duration_ms":68831,"temperature":0.7,"pith_summary":"The paper aims to prove that the two-dimensional contact Muskat problem, in which two fluids in a porous medium are separated by an unknown interface touching the fixed boundary at acute corners and surface tension is absent, has a unique classical solution for a short time. This would be the first local well-posedness result for the two-phase version of the problem in corner domains. If the proof is correct, the two corner points do not move during the existence time and the corner geometry is preserved, a behavior the paper calls the waiting time phenomenon. The main theorem treats angles that are rational multiples of pi, and a second theorem extends the result to arbitrary acute angles by an approximation argument.","feed_headline":"Contact Muskat problem proved locally well-posed at acute corners","feed_subtitle":"Zero-surface-tension interface pinned at acute corners moves smoothly for a short time, with corners fixed.","key_machinery":"The argument is carried by a sequence of reductions. A Hanzawa-type transformation, built from the unknown interface displacement s(omega, tau), fixes the moving boundary and turns problem (1.1) into a nonlinear problem in time-independent domains; linearizing about the initial data produces a transmission problem for the Laplacian with a dynamic boundary condition containing the time derivative of s. The key technical object is a model corner problem in two plane sectors of angle delta, solved via Fourier and Laplace transforms. Solvability reduces to a functional difference equation, equation (4.11), whose coefficient is expressed through the special functions S+(z) = sin(z - theta1) + q2 sin(q1 z - theta2) and S-(z) = sin z - q* sin q1 z; their explicit factorizations and zero locations determine the admissible weight exponent s in the Holder spaces. A continuation argument pastes the corner models together, and the nonlinear problem is closed by a contraction mapping argument.","core_discovery":"The paper's central assertion is Theorem 3.1: under assumptions (h1)-(h7), the reformulated nonlinear problem (3.9) has a unique local classical solution (U1, U2, s) on a short time interval [0, T*], with the pressures in weighted Holder classes $E^{{2+beta,beta,beta}}$_{s+2} and the interface displacement s in $E^{{2+beta,beta,beta}}$_{s+2,s*-1}. The free boundary is reconstructed from s by Gamma(tau) = {m(omega) + l s(omega, tau)}, and the initial pressure distribution solves the stationary transmission problem (3.10). Theorem 6.2 removes the rationality assumption on the angles by taking limits of solutions for approximating rational angles. Corollary 3.1 states that the corner points A0 and A1 remain fixed on [0, T*], giving the waiting-time phenomenon.","pith_inferences":["A natural next step is to test the same machinery with small surface tension; the spectral obstruction encoded in assumption (h7) suggests the admissible weight range may close up as an angle approaches pi/4, hinting that the angle bound may be close to optimal.","The waiting-time conclusion suggests a contact-line pinning mechanism that might persist under perturbations of the initial data within the stated classes, since the solution depends continuously on the data through the contraction construction.","Because Lemma 4.4 is stated without proof, a direct verification of the weighted estimates for the explicit kernels in representation (4.12) would independently confirm the key step; this is a concrete, finite calculation for a fixed rational angle such as delta = pi/6.","The same linear-corner tool set could be applied to the one-phase contact Hele-Shaw problem with time-fractional derivatives, since the functional equation (4.11) has the same structure for those models."],"forward_implications":["The result gives the first local well-posedness theorem for the two-phase contact Muskat problem with zero surface tension and acute corners.","The free boundary preserves its corner geometry on the existence interval: the two corner points are stationary and the interface remains a simple curve meeting the fixed boundary at the same angles.","If the initial normal derivatives of pressure on the interface are negative and the viscosity ratio satisfies k2/k1 < 1, meaning a more viscous fluid is displaced by a less viscous one, the problem is locally well-posed.","The same result holds for arbitrary acute angles in (0, pi/4), not only rational multiples of pi, by the approximation procedure of Section 6.","The model corner problem with a dynamic transmission condition is solvable globally in time in weighted Holder spaces, independently of the nonlinear application, as stated in Theorems 4.1-4.3."],"supporting_citations":[{"why":"Supplies the explicit solution and asymptotic behavior of the functional difference equation (4.11), along with the factorization and zero structure of the special functions S+ and S-.","marker":"[37]"},{"why":"Its Sections 3-6 are cited as the source of the estimates in Lemma 4.4, the bound on the inverse operator that underpins the contraction argument.","marker":"[35]"},{"why":"Provides the solvability and estimates for mixed Dirichlet-transmission problems used for the inhomogeneous right-hand sides in Theorem 4.2 and Lemma 5.1.","marker":"[34]"},{"why":"Proves the one-phase contact problem with a dynamic boundary condition in a corner, used as the epsilon = 0 case in the continuation method of Theorem 5.1.","marker":"[36]"},{"why":"The reduction of the moving-boundary problem to a fixed domain and the model problem in half-planes with a dynamic condition are adapted from this two-phase Hele-Shaw analysis.","marker":"[6]"},{"why":"Introduces the Hanzawa-type transformation that fixes the free boundary, adapted here to the nonsmooth setting.","marker":"[23]"},{"why":"Supplies the approximation technique for irrational angles and the waiting-time setting for the Muskat problem with surface tension.","marker":"[5]"},{"why":"Establishes well-posedness of two-phase Hele-Shaw flow without surface tension under the Rayleigh-Taylor condition, which motivates assumption (h4).","marker":"[3]"}],"fun_headline_variants":["Contact Muskat at acute corners: local well-posedness proved","Acute-corner Muskat: local well-posedness proved","Contact Muskat: local solvability at acute corners","Unique local solution for contact Muskat with acute corners","Waiting time for contact Muskat at acute corners"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the stated a priori estimates for the model corner problem with a dynamic boundary condition, namely Lemma 4.4, whose proof is omitted with a reference to earlier work, actually hold for all data allowed by the assumptions; without those estimates, the bound on the inverse linear operator and the final contraction argument have no basis.","fun_headline_variants_meta":{"raw":{"variants":["Contact Muskat at acute corners: local well-posedness proved","Acute-corner Muskat: local well-posedness proved","Contact Muskat: local solvability at acute corners","Unique local solution for contact Muskat with acute corners","Waiting time for contact Muskat at acute corners"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001296,"raw_usage":{"total_tokens":5219,"prompt_tokens":805,"completion_tokens":4414,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":421,"completion_tokens_details":{"reasoning_tokens":4332}},"tokens_in":421,"tokens_out":4414,"duration_ms":30133,"temperature":1.0,"reasoning_tokens":4332,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:48:07.825902+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the solution of the linear corner problem (4.9) for a fixed rational angle, say delta = pi/6, using the explicit integral representation (4.12), and numerically evaluate the weighted Holder norm of (u1, u2) for a compactly supported smooth f1; if the norm grows faster than the bound in Lemma 4.4 as the time horizon T is increased, or if any nonzero f1 with zero initial data gives a non-unique solution, the central claim collapses.","supporting_citations":[{"cited_title":"Vasylyeva, On a class of functional difference equati ons: explicit solutions, asymptotic behavior and applicat ions, Aequations Mathematicae 98 (2024) 99–171","cited_arxiv_id":null,"evidence_quote":"Supplies the explicit solution and asymptotic behavior of the functional difference equation (4.11), along with the factorization and zero structure of the special functions S+ and S-."},{"cited_title":"Vasylyeva, On the solvability of some nonclassical b oundary-value problem for the Laplace equation in the plane corner, Advan","cited_arxiv_id":null,"evidence_quote":"Its Sections 3-6 are cited as the source of the estimates in Lemma 4.4, the bound on the inverse operator that underpins the contraction argument."},{"cited_title":"Vasylyeva, Existence of smooth solutions of the Hele -Shaw problem in a nonregular domain, Nonlinear Boundary Value Problems 19 (2009) 12–28","cited_arxiv_id":null,"evidence_quote":"Proves the one-phase contact problem with a dynamic boundary condition in a corner, used as the epsilon = 0 case in the continuation method of Theorem 5.1."},{"cited_title":"Bazaliy, N","cited_arxiv_id":null,"evidence_quote":"The reduction of the moving-boundary problem to a fixed domain and the model problem in half-planes with a dynamic condition are adapted from this two-phase Hele-Shaw analysis."},{"cited_title":"Hanzawa, Classical solution of the Stefan problem , Tohoku Math","cited_arxiv_id":null,"evidence_quote":"Introduces the Hanzawa-type transformation that fixes the free boundary, adapted here to the nonsmooth setting."},{"cited_title":"Bazaliy, N","cited_arxiv_id":null,"evidence_quote":"Supplies the approximation technique for irrational angles and the waiting-time setting for the Muskat problem with surface tension."},{"cited_title":"Ambrose, W ell-posedness of two-phase Hele-Shaw flow w ithout surface tension, European J","cited_arxiv_id":null,"evidence_quote":"Establishes well-posedness of two-phase Hele-Shaw flow without surface tension under the Rayleigh-Taylor condition, which motivates assumption (h4)."}],"review_version":1}