{"paper":{"title":"The Energy Based Near Singularity for Fourier Spectral 3D Navier-Stokes Equations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Fourier spectral discretization of the 3D Navier-Stokes equations converges exponentially in space while an energy-based criterion links numerical blowup to loss of regularity.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Beibei Li","submitted_at":"2026-04-25T06:12:04Z","abstract_excerpt":"We investigate the three-dimensional incompressible Navier-Stokes equations. The equations are discretized with Fourier spectral method and a fourth-order Runge-Kutta scheme in time. The spectral accuracy, resolution conditions, and an energy based conditional regularity framework are established analytically. Then we prove exponential convergence, algebraic convergence, and an a posteriori criterion that links numerical blowup to loss of regularity. This work develops a suite of diagnostics for detecting potential finite time singular behavior."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove exponential convergence in space, algebraic convergence in time, and an a posteriori criterion that links numerical blowup to loss of regularity.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The resolution conditions and spectral accuracy assumptions hold uniformly for the chosen Fourier basis and time-stepping scheme when the solution remains in the regularity class under study.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"An energy-based conditional regularity framework and a posteriori diagnostics are derived for Fourier spectral discretizations of the 3D Navier-Stokes equations to detect potential finite-time singular behavior.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Fourier spectral discretization of the 3D Navier-Stokes equations converges exponentially in space while an energy-based criterion links numerical blowup to loss of regularity.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"dbff9f5820d4284948b074632bc4f64c14556f0bb9a75a3b03fb9f55bc569cef"},"source":{"id":"2604.23159","kind":"arxiv","version":2},"verdict":{"id":"167fc5a0-f72d-4385-934b-535d7bf2a369","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T07:45:30.076290Z","strongest_claim":"We prove exponential convergence in space, algebraic convergence in time, and an a posteriori criterion that links numerical blowup to loss of regularity.","one_line_summary":"An energy-based conditional regularity framework and a posteriori diagnostics are derived for Fourier spectral discretizations of the 3D Navier-Stokes equations to detect potential finite-time singular behavior.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The resolution conditions and spectral accuracy assumptions hold uniformly for the chosen Fourier basis and time-stepping scheme when the solution remains in the regularity class under study.","pith_extraction_headline":"Fourier spectral discretization of the 3D Navier-Stokes equations converges exponentially in space while an energy-based criterion links numerical blowup to loss of regularity."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.23159/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T23:24:33.162136Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"38effc70b9c5ccce6ac73bfdafd203c4af5e4829dfa508df13a06bbd50db44a2"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}