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In particular, a finite value of this integral allows smooth continuation past that time."}]},"id":"claim-euler-vorticity-breakdown-criterion","slug":"euler-vorticity-breakdown-criterion","version":1,"kind":"theorem","title":"Finite-time smooth breakdown requires divergent integrated vorticity","assumptions":["Classical divergence-free Euler solution on ℝ³ in H^s with s > 5/2.","The solution is smooth up to each earlier time and the continuation criterion applies."],"scope":"Beale–Kato–Majda is a conditional criterion; it does not itself construct a singularity.","source_entry_ids":["euler-regularity:entry:bkm:v1","euler-regularity:entry:paper:v1"],"formal_declarations":[]}],"problem":{"resolution_scope":{"resolution":"unresolved","extent":"general_case","summary":"OpenAI reports finite-time breakdown from smooth compactly supported unforced Euler data, with a detailed formal theorem. 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Lower-regularity data, bounded domains and forced Euler are different scopes."}]},"aliases":["Unforced Euler blowup","Euler global regularity","Smooth 3D Euler singularity"],"fields":["Partial differential equations","Mathematical physics"],"msc":["35Q31","76B03"],"topics":[],"background":"specialist","scale":"major conjecture","depth":"maintained","source_entry_ids":["euler-regularity:entry:paper:v1","fluid-regularity:entry:announcement:v1","fluid-regularity:entry:lean:v2","euler-regularity:entry:bkm:v1","euler-regularity:entry:lean-singularity:v2","euler-regularity:entry:lean-nonexistence:v2"],"status_report_ids":["euler-regularity:report:singularity:v1"],"editorial":{"collection_id":"openmath-pilot","collection_title":"OpenMath pilot collection","reviewer":"OpenMath source curation","state":"under_assessment","scope":"Source comparison and reader curation; independent mathematical assessment remains pending.","rationale":"OpenAI reports finite-time breakdown from smooth compactly supported unforced Euler data, with a detailed formal theorem. 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In particular, a finite value of this integral allows smooth continuation past that time.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For an unforced three-dimensional Euler solution in the classical Sobolev class "},{"kind":"math","text":"H^s(ℝ³)","latex":"H^s(\\mathbb R^3)"},{"kind":"text","text":", "},{"kind":"math","text":"s > 5/2","latex":"s>5/2"},{"kind":"text","text":", a finite maximal smooth lifespan T* requires "},{"kind":"math","text":"∫₀ᵀ* ‖curl u(t)‖∞ dt = ∞","latex":"\\int_0^{T_*}\\|\\operatorname{curl}u(t)\\|_{\\infty}\\,dt=\\infty"},{"kind":"text","text":". In particular, a finite value of this integral allows smooth continuation past that time."}]},"id":"claim-euler-vorticity-breakdown-criterion","slug":"euler-vorticity-breakdown-criterion","version":1,"kind":"theorem","title":"Finite-time smooth breakdown requires divergent integrated vorticity","assumptions":["Classical divergence-free Euler solution on ℝ³ in H^s with s > 5/2.","The solution is smooth up to each earlier time and the continuation criterion applies."],"scope":"Beale–Kato–Majda is a conditional criterion; it does not itself construct a singularity.","source_entry_ids":["euler-regularity:entry:bkm:v1","euler-regularity:entry:paper:v1"],"formal_declarations":[],"relation":"conditional_implication","relationship_note":"Beale–Kato–Majda is a conditional criterion; it does not itself construct a singularity.","result":"proved","report_ids":["euler-regularity:report:criterion:v1"],"independent_review":"pending","reading":{"placement":"consequence","note":"Beale–Kato–Majda is a conditional criterion; it does not itself construct a singularity."}}],"entries":[{"id":"euler-regularity:entry:paper:v1","source_version_id":"euler-regularity:source:paper:v1","item_id":"paper","url":"https://cdn.openai.com/pdf/315b36cd-ec98-4023-8342-93345194ece1/euler.pdf","wording":"The manuscript states smooth compactly supported unforced Euler data with finite maximal lifespan.","wording_kind":"paraphrase","locator":"Theorem 1.1 and Section 1.1","supersedes":null},{"id":"euler-regularity:entry:bkm:v1","source_version_id":"euler-regularity:source:bkm:v1","item_id":"bkm","url":"https://doi.org/10.1007/BF01212349","wording":"The classical continuation criterion makes integrated supremum vorticity a necessary divergence diagnostic at finite-time smooth breakdown.","wording_kind":"paraphrase","locator":"Communications in Mathematical Physics 94 (1984), 61–66; criterion also recalled in the 2026 manuscript Section 1.1","supersedes":null}],"sources":[{"work":{"authors":["OpenAI"],"kind":"research_paper","publication":"preprint","publication_note":"Source publication and attributed assessments are distinct from independent OpenMath review.","ai_role":"self_reported","ai_note":"OpenAI attributes the argument to its internal research model. Bibliographic authorship is printed as OpenAI; individual human preparation and checking roles are not inferred.","artifacts":[],"published":{"value":"2026-09-08"}},"id":"euler-regularity:source:paper:v1","source_id":"euler-regularity-paper","title":"Finite Time Blowup for the Euler Equation","owner":"OpenAI","url":"https://cdn.openai.com/pdf/315b36cd-ec98-4023-8342-93345194ece1/euler.pdf","subject_scope":"The manuscript states smooth compactly supported unforced Euler data with finite maximal lifespan.","revision":"2026-09-08","content_sha256":"a0c234518e6c489e16996805023eb2e75c00b7c03455f7a3a5be2c124954bfdd","update_method":"Compare the exact source and publish a new immutable curation revision.","identifier_policy":"Stable source identity with exact versioned passages.","observed_on":"2026-10-13","source_updated_on":null,"rights":"metadata-link","license":"Original source license and document rights apply.","terms_url":"https://cdn.openai.com/pdf/315b36cd-ec98-4023-8342-93345194ece1/euler.pdf","attribution":"OpenAI","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90},{"work":{"authors":["J. Thomas Beale","Tosio Kato","Andrew Majda"],"kind":"research_paper","publication":"published","publication_note":"Source publication and attributed assessments are distinct from independent OpenMath review.","ai_role":"not_reported","ai_note":"No specific AI contribution is attributed by this source record.","artifacts":[],"published":{"value":"1984"}},"id":"euler-regularity:source:bkm:v1","source_id":"euler-regularity-bkm","title":"Remarks on the Breakdown of Smooth Solutions for the 3-D Euler Equations","owner":"J. Thomas Beale, Tosio Kato, Andrew Majda","url":"https://doi.org/10.1007/BF01212349","subject_scope":"The classical continuation criterion makes integrated supremum vorticity a necessary divergence diagnostic at finite-time smooth breakdown.","revision":"1984","content_sha256":null,"update_method":"Compare the exact source and publish a new immutable curation revision.","identifier_policy":"Stable source identity with exact versioned passages.","observed_on":"2026-10-13","source_updated_on":null,"rights":"metadata-link","license":"Original source license and document rights apply.","terms_url":"https://doi.org/10.1007/BF01212349","attribution":"J. Thomas Beale, Tosio Kato, Andrew Majda","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90}]}}}