{"paper":{"title":"The 3D incompressible Euler equations with a passive scalar: a road to blow-up?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"nlin.CD","authors_text":"Edriss S. Titi, John D. Gibbon","submitted_at":"2012-11-16T07:15:07Z","abstract_excerpt":"The 3D incompressible Euler equations with a passive scalar $\\theta$ are considered in a smooth domain $\\Omega\\subset \\mathbb{R}^{3}$ with no-normal-flow boundary conditions $\\bu\\cdot\\bhn|_{\\partial\\Omega} = 0$. It is shown that smooth solutions blow up in a finite time if a null (zero) point develops in the vector $\\bB = \\nabla q\\times\\nabla\\theta$, provided $\\bB$ has no null points initially\\,: $\\bom = \\mbox{curl}\\,\\bu$ is the vorticity and $q = \\bom\\cdot\\nabla\\theta$ is a potential vorticity. The presence of the passive scalar concentration $\\theta$ is an essential component of this criteri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3811","kind":"arxiv","version":3},"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"}