{"id":"629bd067-149e-4691-9c4a-53d0e2229599","arxiv_id":"2606.12388","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Short-time well-posedness established for two-phase Muskat problem with surface tension in log-critical regularity without RT condition, via new Schauder estimates for transmission problems in moving domains.","lead":"The paper proves short-time well-posedness for the two-phase Muskat problem with surface tension, allowing different viscosities, arbitrary density contrasts, rigid boundaries, and no Rayleigh-Taylor sign condition. Initial data can be large in a log-critical Hölder class that reaches near the Lipschitz threshold.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the elliptic estimates in moving domains as the key technical step. Because the full manuscript text was not supplied in the query, no concrete flaw in those estimates can be exhibited; the non-finding is therefore honest rather than manufactured.","tokens_in":1848,"tokens_out":238,"duration_ms":17634,"concrete_test":"Re-derive the principal Schauder bound for the transmission operator (the one controlling the third-order term) from the fixed-domain estimates in Part I by treating the domain perturbation as a small Lipschitz graph; verify that the resulting constant remains bounded for viscosity ratios bounded away from 0 and ∞ and for interfaces at positive distance from the rigid walls.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract describes a coherent strategy: new Schauder estimates for the transmission problem in moving domains are combined with contour formulation and time-weighted Hölder estimates to close short-time existence at the log-critical scale without an RT sign condition. No internal inconsistency, hidden assumption, or unsupported step is visible from the given description of the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper proves short-time well-posedness for the two-phase Muskat problem with surface tension, allowing viscosity jumps, arbitrary density contrast, and rigid boundaries without a Rayleigh-Taylor sign condition. Initial data are large in the log-critical class ˙C^{1,log^κ} ∩ H^1 (κ>1), with the interface assumed to remain a graph uniformly separated from the boundaries. The argument derives new Schauder estimates for the elliptic transmission problem in moving domains, combines them with a contour formulation and time-weighted Hölder estimates to control the third-order parabolic mechanism from surface tension and the nonlinear coupling, and obtains existence, uniqueness, smoothing, and stability in arbitrary dimension. The work builds on the Schauder framework of Part I.","tokens_in":1909,"tokens_out":675,"duration_ms":19461,"significance":"If the moving-domain Schauder estimates close at the log-critical scale, the result reaches the natural Lipschitz threshold (up to logarithmic correction) for a genuinely nonlocal quasilinear free-boundary problem without RT sign assumptions or explicit contour dynamics. This extends prior Muskat theory to the full two-phase setting with boundaries and provides a template for handling transmission problems whose coefficients depend on the unknown interface geometry.","major_comments":[{"comment":"§3.2, Theorem 3.4 and the subsequent a-priori estimates: the claimed Schauder bound for the transmission operator in the moving domains appears to rely on the interface remaining a graph with uniform separation; it is not clear from the statement whether the constant remains controlled when the separation approaches the boundary distance, which is load-bearing for the short-time existence argument that must prevent contact.","section":"§3.2, Theorem 3.4"},{"comment":"§4.3, the time-weighted Hölder estimates closing the nonlinear iteration: the third-order parabolic gain from surface tension is identified, but the precise dependence of the constants on the viscosity jump and density contrast is not displayed explicitly; this makes it difficult to verify that the estimates remain uniform when the contrast is arbitrary (including zero).","section":"§4.3"}],"minor_comments":[{"comment":"Notation for the log-critical space ˙C^{1,log^κ} is introduced without an explicit definition of the seminorm; a one-line recall of the definition used in Part I would improve readability.","section":"Introduction"},{"comment":"Figure 1 (schematic of the two-phase domains) lacks labels for the viscosity and density parameters; adding them would clarify the transmission conditions.","section":"Figure 1"},{"comment":"The statement of the main theorem (Theorem 1.1) lists the smallness condition on the initial data only in the abstract; an explicit quantitative bound in the theorem statement would help readers.","section":"Theorem 1.1"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is the second part of a series; the citation to Part I is appropriate, but the journal may wish to confirm that the new estimates are sufficiently independent to merit separate publication."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive assessment and the detailed comments. We address each major comment below and will incorporate clarifications into the revised manuscript.","responses":[{"response":"We agree that the dependence on the separation distance should be stated explicitly. In the revised manuscript, we will clarify in the statement of Theorem 3.4 that the Schauder constant depends on the minimal separation distance δ > 0 between the interface and the fixed boundaries. In the short-time existence proof, the time of existence T is chosen sufficiently small (depending on the initial data and δ) so that the evolved interface remains at least δ/2 away from the boundaries. This ensures the constant stays controlled throughout the existence interval, preventing contact.","revision_made":"yes","referee_comment":"[§3.2, Theorem 3.4] §3.2, Theorem 3.4 and the subsequent a-priori estimates: the claimed Schauder bound for the transmission operator in the moving domains appears to rely on the interface remaining a graph with uniform separation; it is not clear from the statement whether the constant remains controlled when the separation approaches the boundary distance, which is load-bearing for the short-time existence argument that must prevent contact."},{"response":"The estimates in §4.3 are uniform with respect to the viscosity and density contrasts provided the viscosities are positive and bounded. The dependence enters through the coefficients in the elliptic transmission problem, but the surface tension term provides the third-order regularization independently of the density contrast. We will add a remark in §4.3 making this dependence explicit and confirming uniformity for arbitrary fixed contrasts, including the case of zero density contrast.","revision_made":"yes","referee_comment":"[§4.3] §4.3, the time-weighted Hölder estimates closing the nonlinear iteration: the third-order parabolic gain from surface tension is identified, but the precise dependence of the constants on the viscosity jump and density contrast is not displayed explicitly; this makes it difficult to verify that the estimates remain uniform when the contrast is arbitrary (including zero)."}],"tokens_in":1527,"tokens_out":458,"duration_ms":18747,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core advance is the extension of the Part I Schauder framework to the transmission setting with viscosity jumps and fixed boundaries. They recover the normal velocity from an elliptic Darcy problem in the time-dependent domains rather than a closed contour law, then derive adapted estimates that pick up the third-order parabolic effect of surface tension and close the nonlinear terms at the ˙C^{1,log^κ} scale for κ>1. The result covers arbitrary density contrast, large initial data, and arbitrary dimension while keeping the interface a graph away from the walls.\n\nThe strategy is direct: new elliptic estimates plus contour formulation plus time-weighted Hölder norms. Nothing in the outline relies on hidden reductions or previously fitted quantities; the transmission analysis appears independent of the earlier work. The main standing assumption is that the graph condition and separation from boundaries persist for short time, which is the usual local-well-posedness setup.\n\nThe soft spot is the usual one for these papers: the moving-domain estimates are technically heavy, and one would want to verify how the log terms and the time-dependent coefficients are controlled without losing the sharp constants. The abstract gives a coherent plan, but the details matter.\n\nThis is for specialists in free-boundary incompressible flows. It removes a standard restriction and reaches near the Lipschitz threshold in a setting that previous results avoided, so it is worth a serious referee even if the estimates need careful checking.","headline":"This paper gets short-time well-posedness for the two-phase Muskat problem with surface tension at log-critical regularity, without the Rayleigh-Taylor sign condition, by building new Schauder estimates for the transmission problem in moving domains.","tokens_in":2370,"tokens_out":378,"would_cite":false,"duration_ms":12149,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Short-time well-posedness holds for the two-phase Muskat problem with surface tension for large data in the log-critical class without any Rayleigh-Taylor condition.","keywords":["Muskat problem","surface tension","Schauder estimates","two-phase flow","well-posedness","free boundary problems","log-critical regularity"],"falsifier":"An explicit initial graph in the log-critical class for which the solution immediately ceases to be a graph or the elliptic estimates fail would disprove the short-time well-posedness result.","tokens_in":2752,"feed_emoji":"","tokens_out":442,"duration_ms":22920,"temperature":0.7,"pith_summary":"The paper proves short-time existence, uniqueness, smoothing, and stability for the Muskat problem with surface tension when the interface is a graph uniformly separated from rigid boundaries. It allows different viscosities, arbitrary density contrast, and initial data that are large in the space ˙C^{1,log^κ} ∩ H^1 with κ > 1. The argument recovers the normal velocity from a nonlocal elliptic transmission problem in the moving domains and closes the nonlinear estimates with new Schauder-type bounds adapted to the log-critical scale.","feed_headline":"Muskat problem well-posed at log-critical scale without sign condition","feed_subtitle":"Short-time existence holds for large interfaces with viscosity jumps and rigid boundaries in any dimension","key_machinery":"Schauder estimates adapted to the log-critical scale for the elliptic transmission problem in moving domains.","core_discovery":"Sharp Schauder-type estimates adapted to the log-critical scale are derived for the transmission operators generated by the bulk Darcy flow in moving domains; these estimates identify the third-order parabolic mechanism produced by surface tension and control the nonlinear coupling between interface geometry and the elliptic structure, yielding well-posedness for large interfaces in arbitrary dimension.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Two-phase Muskat well-posed at log-critical scale without sign condition","Schauder estimates give log-critical Muskat well-posedness","Muskat well-posedness in log-critical class with surface tension","Well-posedness for Muskat with viscosity jumps at log-critical scale","Log-critical Schauder estimates for two-phase Muskat transmission"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The interface remains a graph uniformly separated from the fixed boundaries throughout the short-time evolution, and the elliptic transmission problem admits the claimed Schauder estimates at the log-critical scale.","fun_headline_variants_meta":{"raw":{"variants":["Two-phase Muskat well-posed at log-critical scale without sign condition","Schauder estimates give log-critical Muskat well-posedness","Muskat well-posedness in log-critical class with surface tension","Well-posedness for Muskat with viscosity jumps at log-critical scale","Log-critical Schauder estimates for two-phase Muskat transmission"]},"model":"grok-4.3","cost_usd":0.006574,"raw_usage":{"total_tokens":3098,"prompt_tokens":722,"num_sources_used":0,"completion_tokens":90,"cost_in_usd_ticks":65737000,"prompt_tokens_details":{"text_tokens":722,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2286,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":722,"tokens_out":90,"duration_ms":15114,"temperature":1.0,"reasoning_tokens":2286,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T08:59:13.176528+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit initial graph in the log-critical class for which the solution immediately ceases to be a graph or the elliptic estimates fail would disprove the short-time well-posedness result.","supporting_citations":[],"review_version":1}