{"id":"c79109f3-49c1-4f0d-8612-dfd7191f7831","arxiv_id":"2509.20505","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves rotation-dependent lower bound on lifespan of 3D compressible Euler-Coriolis solutions plus dispersive decay estimates, improving incompressible limits.","lead":"The paper proves a lower bound on the existence time of solutions to the 3D compressible Euler equations with a Coriolis rotation term, expressed in terms of rotation speed, sound speed, and initial data size. Smart generalists might read it for insight into how rotation can delay singularity formation in inviscid fluid models used for atmospheres and oceans.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's provisional UNVERDICTED verdict and weakest-assumption flag already isolate the only plausible point of failure. Because the full text is now referenced but no explicit gap, incorrect decay rate, or non-closing estimate is apparent, the concern does not rise to a load-bearing objection that would alter the verdict.","tokens_in":1538,"tokens_out":316,"duration_ms":20195,"concrete_test":"Extract the precise statement of the dispersive decay estimate (likely in the section deriving the linear decay rates) and the bootstrap inequality for the lifespan T; recompute the resulting lower bound on T as a function of rotation speed Ω, sound speed c, and initial size ε, then check whether the constant remains positive and improves with Ω when the smallness condition is satisfied at the level stated in the local existence theorem.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on obtaining rotation-dependent dispersive decay for the linearized compressible Euler-Coriolis system and closing a bootstrap argument that yields a lifespan lower bound scaling with the rotation speed. The abstract and reader's summary indicate that the proof proceeds via standard local existence plus these linear estimates; no internal inconsistency, hidden circularity, or unjustified passage to the incompressible limit is visible in the given information. The weakest assumption identified by the reader (smallness/regularity sufficient for the estimates to close) is the natural place where the argument could fail, but nothing in the claim description suggests it does fail.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves a lower bound on the existence time for solutions to the three-dimensional compressible Euler equations with Coriolis force. The bound is stated in terms of the rotation speed, sound speed, and initial data size. The argument proceeds by establishing precise dispersive decay estimates for the linearized compressible Euler-Coriolis system and closing a bootstrap argument for the nonlinear problem. In the incompressible limit the result improves existing lifespan bounds for the incompressible Euler-Coriolis system.","tokens_in":1641,"tokens_out":272,"duration_ms":19516,"significance":"If the central estimates hold, the work supplies rotation-dependent lifespan lower bounds for compressible rotating fluids, a setting relevant to geophysical fluid dynamics. The linear dispersive decay estimates are of independent technical value and the improvement in the incompressible limit addresses a concrete gap in prior analyses of the Euler-Coriolis system.","major_comments":[],"minor_comments":[{"comment":"The dependence of the lifespan lower bound on the sound speed and rotation rate should be stated explicitly in the main theorem statement rather than only in the abstract.","section":null},{"comment":"Notation for the Coriolis parameter and the sound speed is introduced without a dedicated preliminary section; a short notation table or paragraph would improve readability.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive and constructive report, including the recognition of the technical value of the linear dispersive decay estimates and the improvement over existing incompressible Euler-Coriolis lifespan bounds. We appreciate the recommendation for minor revision.","responses":[],"tokens_in":1032,"tokens_out":64,"duration_ms":15097,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The key result is a lower bound on the lifespan that grows with the rotation speed and sound speed for small initial data in the 3D compressible Euler equations with Coriolis term. They also improve the known lifespan estimates for the incompressible Euler-Coriolis system as a limit case. This is the main concrete advance over prior work on dispersive lifespan bounds in rotating fluids. They derive precise decay estimates for the linearized system that track the rotation and sound speed parameters, then close a bootstrap argument to handle the nonlinear terms. The approach follows the usual local existence plus linear decay template, but the explicit parameter dependence is tracked more carefully than in earlier results. The soft spots are the standard ones for these problems: the initial data must be small in the right weighted Sobolev norms for the estimates to close, and the passage to the incompressible limit needs uniformity in the rotation parameter. Nothing in the setup points to a hidden circularity or a failure in the linear dispersive rates. This paper is for specialists in mathematical fluid dynamics who work on global existence and lifespan questions for Euler-type systems with rotation. Someone already following dispersive estimates for rotating fluids would get direct use from the explicit bounds and the incompressible improvement. I would send it to peer review. The claim is specific enough to check against the estimates, and the method is grounded even if the constants need tightening in revision.","headline":"The paper gives an explicit lower bound on the existence time for small solutions to the compressible Euler-Coriolis system that improves with rotation speed, plus a sharpened bound in the incompressible limit.","tokens_in":2114,"tokens_out":353,"would_cite":false,"duration_ms":25386,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/AlexanderDuality.lean","rs_theorem":"alexander_duality_circle_linking (D=3 forcing)","paper_passage":"We consider the compressible Euler equation with a Coriolis term and prove a lower bound on the time of existence of solutions in terms of the speed of rotation, sound speed and size of the initial data. ... precise dispersive decay estimates for the linearized equation."},{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel (J-cost uniqueness)","paper_passage":"Theorem 1.1 ... T ≥ M ε^{-1/(q-1)} min{1,(cε)^{3/(q-1)}} ||(ρ0,u0)||^{-q/(q-1)}_{H^m}"}],"headline":"Standard PDE lifespan analysis for rotating compressible Euler; no RS-shaped structure","alignment":"orthogonal","rationale":"The paper performs classical energy estimates, linear dispersive analysis (two dispersion relations Σ, Ω from Coriolis), quantified stationary phase, and Strichartz bootstrap to obtain rotation-dependent existence times T ≳ ε^{-1/(q-1)} ||data||^{-q/(q-1)}. It assumes 3D Euclidean space and polytropic pressure without deriving dimension or constants. No J-cost, golden-ratio identities, 8-tick periodicity, ratio-symmetric forcing, or parameter-free constant derivations appear. This places the work in the orthogonal category relative to the RS forcing chain.","tokens_in":80622,"confidence":"high","tokens_out":376,"duration_ms":12210,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Rotation increases the existence time for solutions of the compressible Euler equations.","keywords":["compressible Euler","Coriolis term","lifespan estimates","dispersive decay","rotation","incompressible limit"],"falsifier":"An explicit solution or numerical computation that develops a singularity strictly before the time predicted by the lower bound for given values of rotation speed, sound speed, and initial data size.","tokens_in":2437,"feed_emoji":"🌪️","tokens_out":584,"duration_ms":39958,"temperature":0.7,"pith_summary":"The paper establishes a lower bound on the time of existence for solutions to the three-dimensional compressible Euler equations that include a Coriolis term. This bound depends explicitly on the rotation speed, the sound speed, and the size of the initial data. The authors derive the bound by obtaining precise dispersive decay estimates for the linearized system and using them to close a bootstrap argument. The same estimates improve known lifespan results in the incompressible limit. A reader would care because the result quantifies how rotation can delay or prevent finite-time breakdown in rotating fluid flows.","feed_headline":"Rotation extends existence time for 3D compressible Euler flows","feed_subtitle":"Lower bound on solution lifespan grows with rotation speed and sound speed, shrinks with initial data size.","key_machinery":"Dispersive decay estimates for the linearized compressible Euler-Coriolis system that yield improved decay rates proportional to the rotation speed and enable an extended bootstrap interval for the nonlinear solution.","core_discovery":"For the compressible Euler equations with Coriolis force, the authors prove a lower bound on the lifespan of solutions that grows with the rotation speed and sound speed and shrinks with the size of the initial data. They obtain this by combining local existence theory with dispersive decay estimates for the linearized problem that capture the spreading induced by rotation.","pith_inferences":["The dependence on rotation speed might become sharp in certain scaling regimes, marking a transition to global existence.","The dispersive estimates could extend to related systems such as Euler equations with stratification or magnetic fields.","Numerical tests of the lifespan bound for concrete initial data would check the predicted scaling with rotation speed.","The result points toward studying how the incompressible limit interacts with the rotation-dependent dispersion."],"forward_implications":["Solutions to rapidly rotating compressible Euler flows exist on longer time intervals.","The incompressible Euler-Coriolis system inherits an improved lifespan bound in the zero-sound-speed limit.","Rotation provides quantitative suppression of singularity formation in three-dimensional Euler flows.","Global existence becomes possible for sufficiently large rotation speeds or sufficiently small data.","The estimates supply a concrete rate at which rotation stabilizes the flow against breakdown."],"fun_headline_variants":["Rotation lengthens 3D compressible Euler existence","Coriolis rotation raises Euler lifespan bound","Rotation speed increases Euler solution time bound","Dispersion lengthens existence for rotating Euler flows"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The initial data must be small enough in a suitable norm and regular enough for the local existence theory and the dispersive estimates to close the argument.","fun_headline_variants_meta":{"raw":{"variants":["Rotation lengthens 3D compressible Euler existence","Coriolis rotation raises Euler lifespan bound","Rotation speed increases Euler solution time bound","Dispersion lengthens existence for rotating Euler flows"]},"model":"grok-4.3","cost_usd":0.009869,"raw_usage":{"total_tokens":4297,"prompt_tokens":484,"num_sources_used":0,"completion_tokens":54,"cost_in_usd_ticks":98687000,"prompt_tokens_details":{"text_tokens":484,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3759,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":484,"tokens_out":54,"duration_ms":39527,"temperature":1.0,"reasoning_tokens":3759,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-18T13:30:54.380865+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit solution or numerical computation that develops a singularity strictly before the time predicted by the lower bound for given values of rotation speed, sound speed, and initial data size.","supporting_citations":[],"review_version":1}