{"paper":{"title":"Finite-time Singularity Formation for Strong Solutions to the $3D$ Euler equations, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"In-Jee Jeong, Tarek M. Elgindi","submitted_at":"2017-11-08T18:42:13Z","abstract_excerpt":"This work is a companion to [EJE1] and its purpose is threefold: first, we will establish local well-posedness for the axi-symmetric $3D$ Euler equation in the domains $\\{(x_1,x_2,x_3) \\in \\mathbb{R}^3 : x_3^2 \\le \\mathfrak{c}(x_1^2 + x_2^2) \\}$ for $\\mathfrak{c}$ sufficiently small in a scale of critical spaces. Second, we will prove that if the vorticity at $t=0$ can be decomposed into a scale-invariant part and a smoother part vanishing at $x=0$, then this decomposition remains valid for $t>0$ so long as the solution exists. This will then immediately imply singularity formation for finite-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.03089","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"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"}