{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:7MT55NSRRIZXFGNLTYBAIMBEFT","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":"c0a1daac69888b58a5a0e9a8006a9c2811af42a2204fcdc482e2cee58fc8e861","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-05T10:11:06Z","title_canon_sha256":"5088d979c3bea00fa1ecf6cf1deddc0d7bd47086c4afdbfc27b1b0b3c72749ee"},"schema_version":"1.0","source":{"id":"1201.1100","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.1100","created_at":"2026-05-18T04:05:06Z"},{"alias_kind":"arxiv_version","alias_value":"1201.1100v1","created_at":"2026-05-18T04:05:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.1100","created_at":"2026-05-18T04:05:06Z"},{"alias_kind":"pith_short_12","alias_value":"7MT55NSRRIZX","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"7MT55NSRRIZXFGNL","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"7MT55NSR","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:477421c2d317bc35d9aabc1446a4e00954744563f9036bd3ba2c353b6418d8a5","target":"graph","created_at":"2026-05-18T04:05:06Z","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"},"paper":{"abstract_excerpt":"We establish some interior regularity criterions of suitable weak solutions for the 3-D Navier-Stokes equations, which allow the vertical part of the velocity to be large under the local scaling invariant norm. As an application, we improve Ladyzhenskaya-Prodi-Serrin's criterion and Escauriza-Seregin-\\v{S}ver\\'{a}k's criterion. We also show that if weak solution $u$ satisfies $$ \\|u(\\cdot,t)\\|_{L^p}\\leq C(-t)^{\\frac {3-p}{2p}} $$ for some $3<p<\\infty$, then the number of singular points is finite.","authors_text":"Wendong Wang, Zhifei Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-05T10:11:06Z","title":"On the interior regularity criterion and the number of singular points to the Navier-Stokes equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.1100","kind":"arxiv","version":1},"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":"sha256:542811272c0f1755e20f55a8d942b994f95ca8ac2a71d66245af26040e047ef9","target":"record","created_at":"2026-05-18T04:05:06Z","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":"c0a1daac69888b58a5a0e9a8006a9c2811af42a2204fcdc482e2cee58fc8e861","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-05T10:11:06Z","title_canon_sha256":"5088d979c3bea00fa1ecf6cf1deddc0d7bd47086c4afdbfc27b1b0b3c72749ee"},"schema_version":"1.0","source":{"id":"1201.1100","kind":"arxiv","version":1}},"canonical_sha256":"fb27deb6518a337299ab9e020430242cebf071f10312b28dc69f262180d256ea","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fb27deb6518a337299ab9e020430242cebf071f10312b28dc69f262180d256ea","first_computed_at":"2026-05-18T04:05:06.164193Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:05:06.164193Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1dNiCqkC8dj4HcD+jhPAHXKTjHMYdSkUxg34Do3opM18jyJW7xqg3XPadu1YmFxnb1WCPIx2a5cElMwqskhBBA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:05:06.164949Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.1100","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:542811272c0f1755e20f55a8d942b994f95ca8ac2a71d66245af26040e047ef9","sha256:477421c2d317bc35d9aabc1446a4e00954744563f9036bd3ba2c353b6418d8a5"],"state_sha256":"876be5e9cdd559fa7b71e03f00e62cf548463b1d803aea57085128f08e0e9e5a"}