{"paper":{"title":"Exact conservation and the Onsager threshold: a discrete exterior calculus theory for incompressible Navier--Stokes","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"Exact algebraic conservation in a discrete scheme rules out dissipative weak solutions of the Euler equations.","cross_cats":["cs.NA","math.NA"],"primary_cat":"math.AP","authors_text":"Peter Korn","submitted_at":"2026-05-13T06:12:26Z","abstract_excerpt":"We develop a rigorous theory for a structure-preserving discretisation of the incompressible Euler and Navier--Stokes equations, based on discrete exterior calculus on prismatic Delaunay--Voronoi meshes over closed Riemannian manifolds. The central result is a selection principle: exact algebraic conservation at the discrete level is not merely a fidelity property but rules out entire classes of weak solutions that other discretisations reach unconditionally. We establish this in four regimes. \\emph{Smooth solutions}: convergence at rate $\\mathcal{O}(h^{\\min(r_{\\rm rec},\\,r_\\star)}\\,|\\log h|^{"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"exact algebraic conservation at the discrete level is not merely a fidelity property but rules out entire classes of weak solutions that other discretisations reach unconditionally","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"the discrete solutions admit a uniform C^{0,α} bound there (for the inviscid measure-valued regime above the Onsager threshold)","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A structure-preserving DEC discretization of incompressible fluids enforces exact conservation, ruling out dissipative Euler weak solutions and ensuring conservative measure-valued solutions above the Onsager threshold under regularity bounds.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Exact algebraic conservation in a discrete scheme rules out dissipative weak solutions of the Euler equations.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f07d029761e0fe2a30c8fb9aae28652b7f6772a0d86b594405ddfbd5a0bb4d23"},"source":{"id":"2605.13048","kind":"arxiv","version":1},"verdict":{"id":"146baa30-3ba1-4b1c-9610-406539bf6d37","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:37:50.715637Z","strongest_claim":"exact algebraic conservation at the discrete level is not merely a fidelity property but rules out entire classes of weak solutions that other discretisations reach unconditionally","one_line_summary":"A structure-preserving DEC discretization of incompressible fluids enforces exact conservation, ruling out dissipative Euler weak solutions and ensuring conservative measure-valued solutions above the Onsager threshold under regularity bounds.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"the discrete solutions admit a uniform C^{0,α} bound there (for the inviscid measure-valued regime above the Onsager threshold)","pith_extraction_headline":"Exact algebraic conservation in a discrete scheme rules out dissipative weak solutions of the Euler equations."},"references":{"count":55,"sample":[{"doi":"","year":2006,"title":"Finiteelementexteriorcalculus, homological techniques, and applications.Acta Numer., 15:1–155, 2006","work_id":"6d28b46d-c0ae-4ec8-884f-75362c8f9405","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2010,"title":"D. N. Arnold, R. S. Falk, and R. Winther. Finite element exterior calculus: from Hodge theory to numerical stability.Bull. Amer. Math. Soc. (N.S.), 47(2):281–354, 2010. 72","work_id":"1000d048-b10c-43ec-9d64-ffdc1622302e","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1966,"title":"V. I. Arnold. Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits.Annales de l’Institut Fourier, 16(1):319–361, 1966","work_id":"199e9aa3-4fa9-4c5c-9ff0-9807b0f820b7","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2006,"title":"L. C. Berselli, T. Iliescu, and W. J. Layton.Mathematics of Large Eddy Simulation of Turbulent Flows. Scientific Computation. Springer, Berlin, 2006","work_id":"3d4e64fd-0ec2-45e6-8052-6d2087c1a555","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1988,"title":"A. Bossavit. 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