{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:K4VZYBFBIVT2527VHVSNXQDZE2","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c8459d60a7f65d7cbcb5994ca0694c4f6e4d72c1ff29b35d8883a6493e2a19e7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-05-24T14:57:48Z","title_canon_sha256":"b22e2203509113626c21a5c2a9efd38181fab20821da9f4e5ba8a946cbba5432"},"schema_version":"1.0","source":{"id":"2005.11764","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2005.11764","created_at":"2026-07-05T03:59:20Z"},{"alias_kind":"arxiv_version","alias_value":"2005.11764v2","created_at":"2026-07-05T03:59:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2005.11764","created_at":"2026-07-05T03:59:20Z"},{"alias_kind":"pith_short_12","alias_value":"K4VZYBFBIVT2","created_at":"2026-07-05T03:59:20Z"},{"alias_kind":"pith_short_16","alias_value":"K4VZYBFBIVT2527V","created_at":"2026-07-05T03:59:20Z"},{"alias_kind":"pith_short_8","alias_value":"K4VZYBFB","created_at":"2026-07-05T03:59:20Z"}],"graph_snapshots":[{"event_id":"sha256:5df4d6fad6e09a102141a9b58eee1e3e9801332f37e0e1db94940c3f3923b4dc","target":"graph","created_at":"2026-07-05T03:59:20Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2005.11764/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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 ","authors_text":"Erick Schulz, Matthias Kirchhart","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-05-24T14:57:48Z","title":"Div-Curl Problems and $\\mathbf{H}^1$-regular Stream Functions in 3D Lipschitz Domains"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.11764","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1dc31416ee707a714e3ffb0f03c54d7910ee260b87c7c2a1400eb1e489c53179","target":"record","created_at":"2026-07-05T03:59:20Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c8459d60a7f65d7cbcb5994ca0694c4f6e4d72c1ff29b35d8883a6493e2a19e7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-05-24T14:57:48Z","title_canon_sha256":"b22e2203509113626c21a5c2a9efd38181fab20821da9f4e5ba8a946cbba5432"},"schema_version":"1.0","source":{"id":"2005.11764","kind":"arxiv","version":2}},"canonical_sha256":"572b9c04a14567aeebf53d64dbc079268944c4491ed71c77d95ce881b7608cda","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"572b9c04a14567aeebf53d64dbc079268944c4491ed71c77d95ce881b7608cda","first_computed_at":"2026-07-05T03:59:20.561155Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:59:20.561155Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LdRm1sC4NCUk1eYMU+g+3zfKLE1HmlSFxaY4+OcJG3CxSjbWFyWlCnvnbgknoS3iT+jSAP+FnkG6HoQ4gMTPDw==","signature_status":"signed_v1","signed_at":"2026-07-05T03:59:20.561575Z","signed_message":"canonical_sha256_bytes"},"source_id":"2005.11764","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1dc31416ee707a714e3ffb0f03c54d7910ee260b87c7c2a1400eb1e489c53179","sha256:5df4d6fad6e09a102141a9b58eee1e3e9801332f37e0e1db94940c3f3923b4dc"],"state_sha256":"a5f0dbbb9b09a157d183bd10e0899e8815dcd2d74eb12e0740f20a409100725c"}