{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:7YVYDWPJMXK4WSQBZXE6N3BO5G","short_pith_number":"pith:7YVYDWPJ","schema_version":"1.0","canonical_sha256":"fe2b81d9e965d5cb4a01cdc9e6ec2ee99db4e28e68abec7a5915951b843f496c","source":{"kind":"arxiv","id":"2607.01587","version":1},"attestation_state":"computed","paper":{"title":"On the Critical One Components Regularity for the $3-D$ Navier-Stokes System in $L^p_T(\\dot{B}^{\\frac 1 2+\\frac 2 p}_{2,\\infty})$ spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Pei Zheng, Wei Luo, Zhanyang Yin","submitted_at":"2026-07-02T01:39:26Z","abstract_excerpt":"We consider the conditional regularity of the mild solution $v$ of the $3-D$ incompressible Navier-Stokes equations with initial data $v_0\\in \\dot{H}^{\\frac 1 2}$ and vorticity $\\Omega_0\\in L^{r_0}$ for some $r_0\\in (1,2)$. We prove that if the solution associated with initial data $v_0$ blows up at a finite time $T^\\ast$, then for any $2<p<\\infty$, and any unit vectors $e$ in $\\mathbb{R}^3$, the integral\n  $$\\int_0^{T^\\ast}\\left\\Vert (v(t)|e)_{\\mathbb{R}^3}\\right\\Vert_{\\dot{B}^{\\frac 1 2+\\frac 2 p}_{2,\\infty}}^p{\\rm d}t$$ blows up at $T^\\ast$. The conclusion improves the recent results in Che"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.01587","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-07-02T01:39:26Z","cross_cats_sorted":[],"title_canon_sha256":"2009c51ef02aa2f8f1043875ea3b78e857b06169a3e43895b292e9b830abfcef","abstract_canon_sha256":"0e01b4336633dca78191285c741718d473cbe8ecab58762ec93a77c1eebb18df"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T01:17:02.664993Z","signature_b64":"iQA7HS3w1kGiUFZZ/TnEN/55Bwh0jxE1kwrH4IE55QXfF1DN/LCrfZP/pzD9fAvUtkXbRqkYO/TvT8OkRG2PDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe2b81d9e965d5cb4a01cdc9e6ec2ee99db4e28e68abec7a5915951b843f496c","last_reissued_at":"2026-07-03T01:17:02.664495Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T01:17:02.664495Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Critical One Components Regularity for the $3-D$ Navier-Stokes System in $L^p_T(\\dot{B}^{\\frac 1 2+\\frac 2 p}_{2,\\infty})$ spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Pei Zheng, Wei Luo, Zhanyang Yin","submitted_at":"2026-07-02T01:39:26Z","abstract_excerpt":"We consider the conditional regularity of the mild solution $v$ of the $3-D$ incompressible Navier-Stokes equations with initial data $v_0\\in \\dot{H}^{\\frac 1 2}$ and vorticity $\\Omega_0\\in L^{r_0}$ for some $r_0\\in (1,2)$. We prove that if the solution associated with initial data $v_0$ blows up at a finite time $T^\\ast$, then for any $2<p<\\infty$, and any unit vectors $e$ in $\\mathbb{R}^3$, the integral\n  $$\\int_0^{T^\\ast}\\left\\Vert (v(t)|e)_{\\mathbb{R}^3}\\right\\Vert_{\\dot{B}^{\\frac 1 2+\\frac 2 p}_{2,\\infty}}^p{\\rm d}t$$ blows up at $T^\\ast$. The conclusion improves the recent results in Che"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01587","kind":"arxiv","version":1},"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/2607.01587/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.01587","created_at":"2026-07-03T01:17:02.664570+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.01587v1","created_at":"2026-07-03T01:17:02.664570+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01587","created_at":"2026-07-03T01:17:02.664570+00:00"},{"alias_kind":"pith_short_12","alias_value":"7YVYDWPJMXK4","created_at":"2026-07-03T01:17:02.664570+00:00"},{"alias_kind":"pith_short_16","alias_value":"7YVYDWPJMXK4WSQB","created_at":"2026-07-03T01:17:02.664570+00:00"},{"alias_kind":"pith_short_8","alias_value":"7YVYDWPJ","created_at":"2026-07-03T01:17:02.664570+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7YVYDWPJMXK4WSQBZXE6N3BO5G","json":"https://pith.science/pith/7YVYDWPJMXK4WSQBZXE6N3BO5G.json","graph_json":"https://pith.science/api/pith-number/7YVYDWPJMXK4WSQBZXE6N3BO5G/graph.json","events_json":"https://pith.science/api/pith-number/7YVYDWPJMXK4WSQBZXE6N3BO5G/events.json","paper":"https://pith.science/paper/7YVYDWPJ"},"agent_actions":{"view_html":"https://pith.science/pith/7YVYDWPJMXK4WSQBZXE6N3BO5G","download_json":"https://pith.science/pith/7YVYDWPJMXK4WSQBZXE6N3BO5G.json","view_paper":"https://pith.science/paper/7YVYDWPJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.01587&json=true","fetch_graph":"https://pith.science/api/pith-number/7YVYDWPJMXK4WSQBZXE6N3BO5G/graph.json","fetch_events":"https://pith.science/api/pith-number/7YVYDWPJMXK4WSQBZXE6N3BO5G/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7YVYDWPJMXK4WSQBZXE6N3BO5G/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7YVYDWPJMXK4WSQBZXE6N3BO5G/action/storage_attestation","attest_author":"https://pith.science/pith/7YVYDWPJMXK4WSQBZXE6N3BO5G/action/author_attestation","sign_citation":"https://pith.science/pith/7YVYDWPJMXK4WSQBZXE6N3BO5G/action/citation_signature","submit_replication":"https://pith.science/pith/7YVYDWPJMXK4WSQBZXE6N3BO5G/action/replication_record"}},"created_at":"2026-07-03T01:17:02.664570+00:00","updated_at":"2026-07-03T01:17:02.664570+00:00"}