{"id":"6ea17108-033d-47d4-a028-7e96361c38b0","arxiv_id":"2604.10682","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Kernel-adapted Schauder estimates in critical Hölder/Besov spaces yield local and global well-posedness for the Muskat equation with surface tension and Peskin problems with nonlinear elastic tension.","lead":"The paper establishes Schauder-type estimates for linear parabolic systems with variable-coefficient nonlocal operators using a kernel-adapted freezing method, then applies them to prove critical well-posedness for quasilinear nonlocal equations. This framework covers the Muskat equation with surface tension and Peskin problems in 2D and 3D, offering a unified approach for fluid interface models.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the technical hinge (kernel-adapted freezing without lower-order treatment). Because the full manuscript proofs, kernel constructions, and space definitions are not supplied for line-by-line inspection, no concrete counter-example or hidden assumption can be exhibited. The abstract description is internally consistent with standard techniques for nonlocal parabolic Schauder theory, so the UNVERDICTED verdict is left unchanged.","tokens_in":1735,"tokens_out":270,"duration_ms":26184,"concrete_test":"Re-derive the representation formula and residual bound in the linear Schauder estimate (presumably Theorem 1.1 or §3) for a model variable-coefficient operator with the exact kernel decay stated in the paper; check whether the constant remains uniform when the coefficient variation is taken at the critical Hölder/Besov regularity.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The kernel-adapted freezing method is presented as controlling residuals inside the leading-order dynamics via explicit fundamental-kernel representations rather than perturbation arguments. No internal inconsistency appears in the stated assumptions on kernel existence, coefficient regularity, or the critical time-weighted spaces; the applications to Muskat and Peskin equations are described as fitting the resulting abstract well-posedness framework.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper establishes Schauder-type estimates for linear parabolic systems driven by variable-coefficient nonlocal pseudo-differential operators of order s>0. These estimates are formulated in critical time-weighted Hölder/Besov-type spaces and are tailored to quasilinear equations at scaling-critical regularity. A key ingredient is a kernel-adapted freezing-coefficient method: after freezing coefficients at a reference point, explicit representation formulas are derived through the corresponding fundamental kernels and bounds are evaluated at the physical point. This avoids treating coefficient variation as a separate lower-order perturbation. As an application, the paper obtains a general well-posedness framework for a class of nonlocal quasilinear parabolic equations in critical spaces, proving critical local (and in suitable regimes global) well-posedness for the Muskat equation with surface tension and for the two- and three-dimensional Peskin problems with nonlinear elastic tension.","tokens_in":1808,"tokens_out":468,"duration_ms":54041,"significance":"If the estimates hold, the work supplies a unified critical-space well-posedness framework for several distinct nonlocal evolution equations arising in fluid dynamics. The explicit use of fundamental kernels to control residuals inside the leading-order dynamics, rather than via perturbation arguments, is a technical strength that could extend to other quasilinear nonlocal problems. The applications to the Muskat and Peskin models demonstrate concrete utility of the abstract theory.","major_comments":[],"minor_comments":[{"comment":"The abstract states that the kernel-adapted method 'yields robust control of the residual terms within the leading-order dynamics,' but the precise statement of the resulting Schauder estimate (including the dependence on the modulus of continuity of the coefficients) should be displayed as a numbered theorem early in the paper for immediate reference.","section":null},{"comment":"The critical time-weighted Hölder/Besov spaces are central to the claims; an explicit definition or a short paragraph recalling their norms (especially the time-weighting) would improve readability, even if standard in the literature.","section":null},{"comment":"In the applications section, the precise ranges of the order s and the regimes guaranteeing global well-posedness should be stated explicitly rather than described only as 'suitable regimes.'","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive summary of our manuscript and for recommending minor revision. The referee's description correctly identifies the core technical contribution—the kernel-adapted freezing-coefficient method that controls residuals inside the leading-order dynamics rather than as a perturbation—and the applications to the Muskat and Peskin problems. No major comments were listed in the report, so we have no specific points requiring point-by-point rebuttal at this stage. We remain ready to address any minor issues that may arise during the revision process.","responses":[],"tokens_in":1297,"tokens_out":120,"duration_ms":32559,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper's core advance is a kernel-adapted freezing method that produces Schauder estimates for variable-coefficient nonlocal parabolic operators in critical spaces and yields well-posedness for the Muskat and Peskin models. They freeze coefficients at a reference point, pull out explicit fundamental-kernel representations, and evaluate the bounds directly at the physical point. This keeps residual control inside the leading-order terms instead of pushing coefficient variation into a lower-order perturbation. That step looks like the actual technical novelty for scaling-critical nonlocal quasilinear systems. The applications then follow in a straightforward way: local well-posedness in critical time-weighted Hölder/Besov spaces for both models, plus global well-posedness in suitable regimes for the Peskin problems in two and three dimensions. The abstract framework is presented cleanly and the models fit the hypotheses without obvious stretching. The estimates rest on standard kernel assumptions plus the stated regularity on coefficients, with no visible internal contradictions or circular definitions. One soft spot is the dependence on the existence and decay properties of the fundamental kernels for the frozen operators; if those kernels fail to satisfy the required pointwise bounds for the specific nonlocal symbols arising in the Muskat or Peskin settings, the residual estimates may not close at the claimed regularity. The global results are also restricted to suitable regimes, which is stated plainly but narrows the practical reach. This work is for people already working on nonlocal parabolic equations in fluid models and on critical-space techniques. A reader who knows the local Schauder theory and wants to see the nonlocal variable-coefficient extension will find the method and the unified statements useful. It deserves a serious referee to check the kernel derivations and the precise closure of the estimates in the applications.","headline":"The paper's core advance is a kernel-adapted freezing method that produces Schauder estimates for variable-coefficient nonlocal parabolic operators in critical spaces and yields well-posedness for the Muskat and Peskin models.","tokens_in":2315,"tokens_out":430,"would_cite":false,"duration_ms":51706,"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":"A kernel-adapted freezing method produces Schauder estimates that establish critical well-posedness for nonlocal quasilinear fluid equations.","keywords":["Schauder estimates","nonlocal parabolic equations","quasilinear equations","Muskat equation","Peskin problem","critical spaces","well-posedness","fluid dynamics"],"falsifier":"An explicit example of a variable-coefficient nonlocal pseudo-differential operator satisfying the kernel and regularity hypotheses for which the Schauder estimates fail to hold in the critical spaces, or a demonstration that the Muskat equation with surface tension lacks local well-posedness at the scaling-critical regularity.","tokens_in":2636,"feed_emoji":"","tokens_out":787,"duration_ms":26262,"temperature":0.7,"pith_summary":"The paper develops Schauder-type estimates for linear parabolic systems driven by variable-coefficient nonlocal pseudo-differential operators, formulated in critical time-weighted Hölder and Besov spaces. A kernel-adapted freezing-coefficient method derives explicit representation formulas from fundamental kernels after freezing at a reference point, then evaluates bounds at the physical point to control residuals from coefficient variation directly inside the leading-order dynamics. This yields a general well-posedness framework for a class of nonlocal quasilinear parabolic equations at scaling-critical regularity. As concrete applications, the results give critical local well-posedness for the Muskat equation with surface tension and critical local and global well-posedness in suitable regimes for the two- and three-dimensional Peskin problems with nonlinear elastic tension.","feed_headline":"Kernel freezing controls residuals in nonlocal parabolic estimates","feed_subtitle":"Schauder estimates in critical spaces give local and global well-posedness for the Muskat and Peskin equations.","key_machinery":"The kernel-adapted freezing-coefficient method, which freezes coefficients at a reference point to obtain explicit representation formulas via fundamental kernels and controls residual terms from coefficient variation inside the leading-order dynamics.","core_discovery":"We establish Schauder-type estimates for linear parabolic systems driven by variable-coefficient nonlocal pseudo-differential operators of order s>0. These estimates are formulated in critical time-weighted Hölder/Besov-type spaces and are tailored to quasilinear equations at scaling-critical regularity. A key ingredient is a kernel-adapted freezing-coefficient method. After freezing the coefficients at a reference point, we derive explicit representation formulas through the corresponding fundamental kernels and then evaluate the resulting bounds at the physical point. This avoids treating the coefficient variation as a separate lower-order perturbation and yields robust control of the残差项s.","pith_inferences":["The same freezing technique may serve as a template for establishing critical well-posedness in other quasilinear nonlocal models that possess comparable fundamental kernels.","The critical-space results open the possibility of tracking long-time behavior or detecting singularity formation in the Muskat and Peskin problems under the tension terms.","Because residuals are absorbed into the leading dynamics rather than treated as perturbations, the method may simplify analysis of related nonlocal systems in fluid dynamics and adjacent fields."],"forward_implications":["Critical local well-posedness holds for the Muskat equation with surface tension in the indicated spaces.","Critical local well-posedness and, in suitable regimes, global well-posedness hold for the two- and three-dimensional Peskin problems with nonlinear elastic tension.","A unified critical framework applies to a class of distinct nonlocal quasilinear parabolic evolution equations arising in fluid dynamics."],"fun_headline_variants":["Kernel freezing for Schauder estimates in nonlocal equations","Critical Schauder bounds for variable-coefficient nonlocal PDEs","Well-posedness of Muskat and Peskin via critical estimates","Nonlocal quasilinear well-posedness in critical Hölder spaces"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The nonlocal operators must admit suitable fundamental kernels and the coefficients must possess the regularity required for the critical time-weighted Hölder/Besov spaces so that residuals from freezing stay controllable within the leading-order terms.","fun_headline_variants_meta":{"raw":{"variants":["Kernel freezing for Schauder estimates in nonlocal equations","Critical Schauder bounds for variable-coefficient nonlocal PDEs","Well-posedness of Muskat and Peskin via critical estimates","Nonlocal quasilinear well-posedness in critical Hölder spaces"]},"model":"grok-4.3","cost_usd":0.006463,"raw_usage":{"total_tokens":2957,"prompt_tokens":690,"num_sources_used":0,"completion_tokens":69,"cost_in_usd_ticks":64628000,"prompt_tokens_details":{"text_tokens":690,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2198,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":690,"tokens_out":69,"duration_ms":36963,"temperature":1.0,"reasoning_tokens":2198,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-10T16:19:59.877955+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit example of a variable-coefficient nonlocal pseudo-differential operator satisfying the kernel and regularity hypotheses for which the Schauder estimates fail to hold in the critical spaces, or a demonstration that the Muskat equation with surface tension lacks local well-posedness at the scaling-critical regularity.","supporting_citations":[],"review_version":1}