{"paper":{"title":"Singularity formation for the two-dimensional harmonic map flow into $S^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Juan Davila, Juncheng Wei, Manuel del Pino","submitted_at":"2017-02-19T21:54:57Z","abstract_excerpt":"We construct finite time blow-up solutions to the 2-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 &= \\varphi \\quad \\text{on } \\partial \\Omega\\times(0,T) \\\\ u(\\cdot,0) &= u_0 \\quad \\text{in } \\Omega , \\end{align*} where $\\Omega$ is a bounded, smooth domain in $\\mathbb{R}^2$, $u: \\Omega\\times(0,T)\\to S^2$, $u_0:\\bar\\Omega \\to S^2$ is smooth, and $\\varphi = u_0\\big|_{\\partial\\Omega}$. Given any points $q_1,\\ldots, q_k$ in the domain, we find initial and boundary data so that the solution blows-up preci"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.05801","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"}