{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:NGYV33PKCZSTHKLW44ATU5AK6R","short_pith_number":"pith:NGYV33PK","schema_version":"1.0","canonical_sha256":"69b15dedea166533a976e7013a740af45bec54babfc7dda1b0bd26e38e8a64d6","source":{"kind":"arxiv","id":"1805.02324","version":1},"attestation_state":"computed","paper":{"title":"Regularity of Solutions of the Camassa-Holm Equations with Fractional Laplacian Viscosity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Linghui Meng, Yong He, Yue Wang, Zaihui Gan","submitted_at":"2018-05-07T02:45:16Z","abstract_excerpt":"We study the existence, uniqueness and regularity of solutions to the $n$-dimensional ($n=2,3$) Camassa-Holm equations with fractional Laplacian viscosity with smooth initial data. It is a coupled system between the Navier-Stokes equations with nonlocal viscosity and a Helmholtz equation. The main difficulty lies in establishing some a priori estimates for the fractional Laplacian viscosity. To achieve this, we need to explore suitable fractional-power Sobolev-type estimates, and bilinear estimates for fractional derivatives. Especially, for the critical case $\\displaystyle s=\\frac{n}{4}$ with"},"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":"1805.02324","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-05-07T02:45:16Z","cross_cats_sorted":[],"title_canon_sha256":"84beec53cbaee696d3216c2941c094c78e0d9c9eed61291cd247847da178f538","abstract_canon_sha256":"2dfd35faa4c5937161a12d3d7b932de698fe8263458c183e7102902c488768ff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:40.170022Z","signature_b64":"tzqZh5cdXI7tbATLzIMGCzxMo1NGstzMwH1Yu4WbkrsSMZS+Oeh8LdxgS3XfbuHs9LxcZls1FHCPG6DojfG9DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"69b15dedea166533a976e7013a740af45bec54babfc7dda1b0bd26e38e8a64d6","last_reissued_at":"2026-05-18T00:16:40.169569Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:40.169569Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Regularity of Solutions of the Camassa-Holm Equations with Fractional Laplacian Viscosity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Linghui Meng, Yong He, Yue Wang, Zaihui Gan","submitted_at":"2018-05-07T02:45:16Z","abstract_excerpt":"We study the existence, uniqueness and regularity of solutions to the $n$-dimensional ($n=2,3$) Camassa-Holm equations with fractional Laplacian viscosity with smooth initial data. It is a coupled system between the Navier-Stokes equations with nonlocal viscosity and a Helmholtz equation. The main difficulty lies in establishing some a priori estimates for the fractional Laplacian viscosity. To achieve this, we need to explore suitable fractional-power Sobolev-type estimates, and bilinear estimates for fractional derivatives. Especially, for the critical case $\\displaystyle s=\\frac{n}{4}$ with"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.02324","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":""},"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":"1805.02324","created_at":"2026-05-18T00:16:40.169638+00:00"},{"alias_kind":"arxiv_version","alias_value":"1805.02324v1","created_at":"2026-05-18T00:16:40.169638+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.02324","created_at":"2026-05-18T00:16:40.169638+00:00"},{"alias_kind":"pith_short_12","alias_value":"NGYV33PKCZST","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_16","alias_value":"NGYV33PKCZSTHKLW","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_8","alias_value":"NGYV33PK","created_at":"2026-05-18T12:32:40.477152+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/NGYV33PKCZSTHKLW44ATU5AK6R","json":"https://pith.science/pith/NGYV33PKCZSTHKLW44ATU5AK6R.json","graph_json":"https://pith.science/api/pith-number/NGYV33PKCZSTHKLW44ATU5AK6R/graph.json","events_json":"https://pith.science/api/pith-number/NGYV33PKCZSTHKLW44ATU5AK6R/events.json","paper":"https://pith.science/paper/NGYV33PK"},"agent_actions":{"view_html":"https://pith.science/pith/NGYV33PKCZSTHKLW44ATU5AK6R","download_json":"https://pith.science/pith/NGYV33PKCZSTHKLW44ATU5AK6R.json","view_paper":"https://pith.science/paper/NGYV33PK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1805.02324&json=true","fetch_graph":"https://pith.science/api/pith-number/NGYV33PKCZSTHKLW44ATU5AK6R/graph.json","fetch_events":"https://pith.science/api/pith-number/NGYV33PKCZSTHKLW44ATU5AK6R/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NGYV33PKCZSTHKLW44ATU5AK6R/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NGYV33PKCZSTHKLW44ATU5AK6R/action/storage_attestation","attest_author":"https://pith.science/pith/NGYV33PKCZSTHKLW44ATU5AK6R/action/author_attestation","sign_citation":"https://pith.science/pith/NGYV33PKCZSTHKLW44ATU5AK6R/action/citation_signature","submit_replication":"https://pith.science/pith/NGYV33PKCZSTHKLW44ATU5AK6R/action/replication_record"}},"created_at":"2026-05-18T00:16:40.169638+00:00","updated_at":"2026-05-18T00:16:40.169638+00:00"}