{"paper":{"title":"Asymptotically Self-Similar Blowup for 3D Incompressible Euler with $C^{1, 1/3-}$ Velocity I: $C^{\\infty}$ 1D Limiting Profiles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For the critical case α=1/3 a C∞ self-similar blowup profile with unbounded stream function is constructed for a 1D model of 3D axisymmetric Euler.","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jiajie Chen","submitted_at":"2026-05-14T17:50:19Z","abstract_excerpt":"We consider a one-parameter family of 1D models for the 3D axisymmetric incompressible Euler equation with $C^{\\alpha}$ vorticity and without swirl near the symmetry axis. For $\\alpha = \\frac13$, we impose a crucial normalization and construct a $C^{\\infty}$ self-similar blowup profile with unbounded 1D stream function and infinite spatial blowup rate, using a fixed-point argument around a numerically constructed approximate profile. For $\\alpha < \\frac13$ sufficiently close to $\\frac13$, we perturb the $\\frac13$-profile and analytically construct exact smooth 1D profiles with bounded stream f"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For α = 1/3 we impose a crucial normalization and construct a C^∞ self-similar blowup profile with unbounded 1D stream function and infinite spatial blowup rate, using a fixed-point argument around a numerically constructed approximate profile.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The numerically constructed approximate profile is sufficiently close to an exact solution so that the fixed-point argument converges in the chosen function space; the abstract does not quantify the approximation error or the contraction constant.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Constructs C^∞ self-similar blowup profiles for 1D models of 3D Euler at α=1/3 using fixed-point around a numerical approximation, plus nearby exact profiles for α slightly below 1/3.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For the critical case α=1/3 a C∞ self-similar blowup profile with unbounded stream function is constructed for a 1D model of 3D axisymmetric Euler.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"1dc7c009ea44f4689f1eab657e3ab2afa1495fd557d1ba451700fcd8f9c85081"},"source":{"id":"2605.15149","kind":"arxiv","version":1},"verdict":{"id":"471e5fbd-d322-44f0-919a-d423ac85ca90","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:57:11.234201Z","strongest_claim":"For α = 1/3 we impose a crucial normalization and construct a C^∞ self-similar blowup profile with unbounded 1D stream function and infinite spatial blowup rate, using a fixed-point argument around a numerically constructed approximate profile.","one_line_summary":"Constructs C^∞ self-similar blowup profiles for 1D models of 3D Euler at α=1/3 using fixed-point around a numerical approximation, plus nearby exact profiles for α slightly below 1/3.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The numerically constructed approximate profile is sufficiently close to an exact solution so that the fixed-point argument converges in the chosen function space; the abstract does not quantify the approximation error or the contraction constant.","pith_extraction_headline":"For the critical case α=1/3 a C∞ self-similar blowup profile with unbounded stream function is constructed for a 1D model of 3D axisymmetric Euler."},"references":{"count":46,"sample":[{"doi":"","year":2008,"title":"An introduction to numerical analysis","work_id":"d479409f-4e6a-4d40-a850-5e9d468a4346","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Smooth imploding solutions for 3D com- pressible fluids","work_id":"67bc8818-7d18-4c07-bcb6-09a03dc8af7e","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Blowup for the defocusing septic complex-valued nonlinear wave equation inR 4+1.To appear in Commun","work_id":"131ea7a3-2e93-4884-8bcc-98137aa819ec","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"Global smooth solutions for the inviscid SQG equation","work_id":"459d01c5-06ee-4b9a-a025-c9d29b80ab34","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"Singularity formation and global well-pos edness for the generalized Constantin–Lax–Majda equation with dissipation","work_id":"e749853d-547a-463f-a71c-ede946d58fd3","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":46,"snapshot_sha256":"818987e441a965cdee9dd0f4f091789f3076d48f7cfda07e3332e02e8a3ad0e5","internal_anchors":1},"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"}