{"paper":{"title":"Asymptotically Self-Similar Blowup for 3D Incompressible Euler with $C^{1, 1/3-}$ Velocity II: 3D Profiles, Blowup, and Limiting behavior","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The 3D incompressible Euler equations without swirl admit exact C^{1,α} self-similar blowup profiles for every α below 1/3, which are reached asymptotically from C_c^α initial vorticity.","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jiajie Chen","submitted_at":"2026-05-14T17:39:47Z","abstract_excerpt":"For any $\\alpha \\in (0, 1/3)$, we construct exact $C^{1,\\alpha}$ self-similar blowup profiles for the 3D incompressible Euler equation without swirl, and build on them to prove asymptotically self-similar blowup from $C_c^\\alpha$ initial vorticity and $C^{1,\\alpha}\\cap L^2$ initial velocity. Moreover, we provide a complete characterization of the limiting behavior of the $C^{1,\\alpha}$ blowup profiles and the associated $C^{1,\\alpha}$ blowup solutions as $\\alpha\\to(1/3)^-$. Specifically, as $\\alpha \\to(1/3)^-$, the spatial blowup rate $c_{x,\\alpha}$ diverges to $\\infty$, while the $C^{1,\\alpha"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For any α ∈ (0, 1/3), we construct exact C^{1,α} self-similar blowup profiles for the 3D incompressible Euler equation without swirl, and build on them to prove asymptotically self-similar blowup from C_c^α initial vorticity and C^{1,α}∩L^2 initial velocity. Moreover, as α→(1/3)^-, the spatial blowup rate diverges while the profile converges to a nonzero multiple of r^{1/3} W̄_{1/3}(z).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The fixed-point map defined from the 1D approximate profile is a contraction in the chosen anisotropic weighted space, and the subsequent finite-codimension stability holds in the low-regularity C^{1,α} topology; both rely on the integration-by-parts identity along trajectories that exploits the Euler equation twice.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Constructs C^{1,α} self-similar blowup profiles for 3D Euler without swirl for α<1/3 and proves asymptotically self-similar blowup with limiting factorization to a 1D profile as α approaches 1/3 from below.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The 3D incompressible Euler equations without swirl admit exact C^{1,α} self-similar blowup profiles for every α below 1/3, which are reached asymptotically from C_c^α initial vorticity.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0d727c6e260e59eae91fe838108a40d7a65a2131d2ceccbbed154dc16d383345"},"source":{"id":"2605.15130","kind":"arxiv","version":1},"verdict":{"id":"f18caab9-e257-43f5-9560-5f7ae6720add","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T03:07:51.749800Z","strongest_claim":"For any α ∈ (0, 1/3), we construct exact C^{1,α} self-similar blowup profiles for the 3D incompressible Euler equation without swirl, and build on them to prove asymptotically self-similar blowup from C_c^α initial vorticity and C^{1,α}∩L^2 initial velocity. Moreover, as α→(1/3)^-, the spatial blowup rate diverges while the profile converges to a nonzero multiple of r^{1/3} W̄_{1/3}(z).","one_line_summary":"Constructs C^{1,α} self-similar blowup profiles for 3D Euler without swirl for α<1/3 and proves asymptotically self-similar blowup with limiting factorization to a 1D profile as α approaches 1/3 from below.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The fixed-point map defined from the 1D approximate profile is a contraction in the chosen anisotropic weighted space, and the subsequent finite-codimension stability holds in the low-regularity C^{1,α} topology; both rely on the integration-by-parts identity along trajectories that exploits the Euler equation twice.","pith_extraction_headline":"The 3D incompressible Euler equations without swirl admit exact C^{1,α} self-similar blowup profiles for every α below 1/3, which are reached asymptotically from C_c^α initial vorticity."},"references":{"count":61,"sample":[{"doi":"","year":2026,"title":"Finite time singularities in the Landau equation with very hard potentials","work_id":"1c92e033-f880-43b0-82b0-d4f76c09c128","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Smooth imploding solutions for 3D com- pressible fluids","work_id":"dbe45c2b-dc5c-4a39-a9fa-f5bde6d80817","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"American Mathematical Soc., 2018","work_id":"39f51527-d69a-4bad-8299-62a3e740c03b","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"Bojin Chen, De Huang, and Xiangyuan Li. Novel self-similar finite-time blowups with singular profiles of the 1D Hou-Luo model and the 2D Boussinesq equations: A numerical investigation.arXiv preprint ","work_id":"6d094b14-05ae-406b-8adf-c0b96efee643","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"Singularity formation and global well-posedness for the generalized Constantin–Lax–Majda equation with dissipation.Nonlinearity, 33(5):2502, 2020","work_id":"5a13357f-3bc5-4775-86e3-7636d6ecf421","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":61,"snapshot_sha256":"6a17ee9cd3d63fbbeb3f5b2a2f865c42c5112fa25c29cdead2bb6df8bea22bf8","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"d9736464403b33d26e7ed3bcde1086a7ba5afd8befca0756287b8e466a361ee2"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}