{"paper":{"title":"Blow-up for the 3-dimensional axially symmetric harmonic map flow into S2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Catalina Pesce, Juan Davila, Juncheng Wei, Manuel del Pino","submitted_at":"2019-02-11T16:53:31Z","abstract_excerpt":"We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere $S^2$, \\begin{align*} u_t & = \\Delta u + |\\nabla u|^2 u \\quad \\text{in } \\Omega\\times(0,T) \\\\ u &= u_b \\quad \\text{on } \\partial \\Omega\\times(0,T) \\\\ u(\\cdot,0) &= u_0 \\quad \\text{in } \\Omega , \\end{align*} with $u(x,t): \\bar \\Omega\\times [0,T) \\to S^2$. Here $\\Omega$ is a bounded, smooth axially symmetric domain in $\\mathbb{R}^3$. We prove that for any circle $\\Gamma \\subset \\Omega$ with the same axial symmetry, and any sufficiently small $T>0$ there exist initial and boundary conditions such tha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.03995","kind":"arxiv","version":1},"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"}