{"paper":{"title":"Well-posedness of a generalized Stokes operator on domains with cylindrical ends via layer-potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Under positivity assumptions on V and V0 the generalized Stokes operator Ξ and its layer potentials become invertible on domains with cylindrical ends.","cross_cats":["math-ph","math.DG","math.FA","math.MP"],"primary_cat":"math.AP","authors_text":"Mirela Kohr, Victor Nistor, Wolfgang Wendland","submitted_at":"2026-05-11T16:59:02Z","abstract_excerpt":"We study the \\emph{generalized Stokes operator} \\begin{equation*} \\bsXi \\ede \\bsXi _{V,V_0} \\ede \\left(\\begin{array}{ccc} \\bsL + V & \\nabla \\\\ \\nabla^* & -V_0 \\end{array}\\right) \\end{equation*} on a \\emph{domain with straight cylindrical ends} $\\Omega$ using \\emph{the method of layer potentials} on $M \\supset \\Omega$. The operator $\\bsXi_{0, 0}$ is the classical Stokes operator. Under suitable positivity assumptions on $V$ and $V_{0}$, we prove that $\\bsXi$ is Fredholm. This allows us then to define the single- and double-layer potentials $\\bsS$ and $\\frac12 + \\bsK$. Under further positivity a"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Under slightly stronger assumptions on V and V0, we prove the invertibility of the operators Ξ, S, and ½ + K. The invertibility of these operators leads to well-posedness results for the associated (linear) Stokes boundary value problem with Dirichlet boundary conditions on Ω. As an application, we prove the well-posedness result for the Dirichlet problem for the generalized Navier-Stokes system with small data on a domain with cylindrical ends.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Under suitable positivity assumptions on V and V0, we prove that Ξ is Fredholm. Under further positivity assumptions, we prove that S and ½ + K are also Fredholm. Under slightly stronger assumptions on V and V0, we prove the invertibility...","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A generalized Stokes operator on cylindrical-end domains is Fredholm and invertible under positivity assumptions on V and V0 via layer potentials, yielding well-posedness for linear Stokes and small-data Navier-Stokes Dirichlet problems.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Under positivity assumptions on V and V0 the generalized Stokes operator Ξ and its layer potentials become invertible on domains with cylindrical ends.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"6f2f98a558793cd825a7f2d1d239379e28eaff23679dc5a686b4d350edb4e3d6"},"source":{"id":"2605.10849","kind":"arxiv","version":2},"verdict":{"id":"de8ca2a0-b717-4cc9-a071-56372da95224","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T03:43:16.473989Z","strongest_claim":"Under slightly stronger assumptions on V and V0, we prove the invertibility of the operators Ξ, S, and ½ + K. The invertibility of these operators leads to well-posedness results for the associated (linear) Stokes boundary value problem with Dirichlet boundary conditions on Ω. As an application, we prove the well-posedness result for the Dirichlet problem for the generalized Navier-Stokes system with small data on a domain with cylindrical ends.","one_line_summary":"A generalized Stokes operator on cylindrical-end domains is Fredholm and invertible under positivity assumptions on V and V0 via layer potentials, yielding well-posedness for linear Stokes and small-data Navier-Stokes Dirichlet problems.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Under suitable positivity assumptions on V and V0, we prove that Ξ is Fredholm. Under further positivity assumptions, we prove that S and ½ + K are also Fredholm. Under slightly stronger assumptions on V and V0, we prove the invertibility...","pith_extraction_headline":"Under positivity assumptions on V and V0 the generalized Stokes operator Ξ and its layer potentials become invertible on domains with cylindrical ends."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.10849/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T05:22:00.321663Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T14:34:20.753185Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T10:31:17.497539Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T08:56:44.592507Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"2d547503c69894b9295c521c094b31fa34b1b60b13c8e678c188f7eb1cb88e9e"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"36a59dbfb1663654f9192ababe4118ca90a3b5d9d0b715ea23e9383830723206"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}