{"paper":{"title":"Div-Curl Problems and $\\mathbf{H}^1$-regular Stream Functions in 3D Lipschitz Domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Erick Schulz, Matthias Kirchhart","submitted_at":"2020-05-24T14:57:48Z","abstract_excerpt":"We consider the problem of recovering the divergence-free velocity field ${\\mathbf U}\\in\\mathbf{L}^2(\\Omega)$ of a given vorticity ${\\mathbf F}=\\mathrm{curl}\\,{\\mathbf U}$ on a bounded Lipschitz domain $\\Omega\\subset\\mathbb{R}^3$. To that end, we solve the \"div-curl problem\" for a given ${\\mathbf F}\\in{\\mathbf H}^{-1}(\\Omega)$. The solution is expressed in terms of a vector potential (or stream function) ${\\mathbf A}\\in{\\mathbf H}^1(\\Omega)$ such that ${\\mathbf U}=\\mathrm{curl}\\,{\\mathbf A}$. After discussing existence and uniqueness of solutions and associated vector potentials, we propose a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.11764","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2005.11764/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}