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More precisely, let $(u,\\tau)$ be the global small solutions constructed in [18],\n  we prove for any $(u_0,\\tau_0)\\in{\\dot{B}_{2,1}^{-s}}(\\mathbb{R}^n)$ that \\begin{eqnarray*} \\big\\|\\Lambda^{\\alpha}(u,\\Lambda^{-1}\\mathbb{P}\\mathrm{div}\\tau)\\big\\|_{L^q} \\le C (1+t)^{-\\frac n4-\\frac {(\\alpha+s)q-n}{2q}}, \\quad\\Lambda\\stackrel{\\mathrm{def}}{=}\\sqrt{-\\Delta}, \\end{e"},"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":"1905.02604","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-05-07T14:21:49Z","cross_cats_sorted":[],"title_canon_sha256":"684e23ca679e31353874fb748ed5a460928936e30b735ce099fb0e572fe2bb98","abstract_canon_sha256":"d1e53777d346023c6748667783e902778e1b4d0142fa5886ee0d03ba29b48ea3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:46:50.938512Z","signature_b64":"BkCvETyDd4aTY8bVdktPqDA4YYS4SzhlKHpRLxsQ2Ht3BaFtqOgAHy1Hy2jbPVn7VhatG+qOKZdXXfqTzV5fDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8929a61e9d80d61fd3507ad4a2573894a249f5e09525e5868a373465519560ed","last_reissued_at":"2026-05-17T23:46:50.938080Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:46:50.938080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal decay for the $n$-dimensional incompressible Oldroyd-B model without damping mechanism","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xiaoping Zhai","submitted_at":"2019-05-07T14:21:49Z","abstract_excerpt":"By a new energy approach involved in the high frequencies and low frequencies decomposition in the Besov spaces, we obtain the optimal decay for the incompressible Oldroyd-B model without damping mechanism in $\\mathbb{R}^n$ ($n\\ge 2$). More precisely, let $(u,\\tau)$ be the global small solutions constructed in [18],\n  we prove for any $(u_0,\\tau_0)\\in{\\dot{B}_{2,1}^{-s}}(\\mathbb{R}^n)$ that \\begin{eqnarray*} \\big\\|\\Lambda^{\\alpha}(u,\\Lambda^{-1}\\mathbb{P}\\mathrm{div}\\tau)\\big\\|_{L^q} \\le C (1+t)^{-\\frac n4-\\frac {(\\alpha+s)q-n}{2q}}, \\quad\\Lambda\\stackrel{\\mathrm{def}}{=}\\sqrt{-\\Delta}, \\end{e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.02604","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":"1905.02604","created_at":"2026-05-17T23:46:50.938144+00:00"},{"alias_kind":"arxiv_version","alias_value":"1905.02604v1","created_at":"2026-05-17T23:46:50.938144+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.02604","created_at":"2026-05-17T23:46:50.938144+00:00"},{"alias_kind":"pith_short_12","alias_value":"REU2MHU5QDLB","created_at":"2026-05-18T12:33:27.125529+00:00"},{"alias_kind":"pith_short_16","alias_value":"REU2MHU5QDLB7U2Q","created_at":"2026-05-18T12:33:27.125529+00:00"},{"alias_kind":"pith_short_8","alias_value":"REU2MHU5","created_at":"2026-05-18T12:33:27.125529+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/REU2MHU5QDLB7U2QPLKKEVZYSS","json":"https://pith.science/pith/REU2MHU5QDLB7U2QPLKKEVZYSS.json","graph_json":"https://pith.science/api/pith-number/REU2MHU5QDLB7U2QPLKKEVZYSS/graph.json","events_json":"https://pith.science/api/pith-number/REU2MHU5QDLB7U2QPLKKEVZYSS/events.json","paper":"https://pith.science/paper/REU2MHU5"},"agent_actions":{"view_html":"https://pith.science/pith/REU2MHU5QDLB7U2QPLKKEVZYSS","download_json":"https://pith.science/pith/REU2MHU5QDLB7U2QPLKKEVZYSS.json","view_paper":"https://pith.science/paper/REU2MHU5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1905.02604&json=true","fetch_graph":"https://pith.science/api/pith-number/REU2MHU5QDLB7U2QPLKKEVZYSS/graph.json","fetch_events":"https://pith.science/api/pith-number/REU2MHU5QDLB7U2QPLKKEVZYSS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/REU2MHU5QDLB7U2QPLKKEVZYSS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/REU2MHU5QDLB7U2QPLKKEVZYSS/action/storage_attestation","attest_author":"https://pith.science/pith/REU2MHU5QDLB7U2QPLKKEVZYSS/action/author_attestation","sign_citation":"https://pith.science/pith/REU2MHU5QDLB7U2QPLKKEVZYSS/action/citation_signature","submit_replication":"https://pith.science/pith/REU2MHU5QDLB7U2QPLKKEVZYSS/action/replication_record"}},"created_at":"2026-05-17T23:46:50.938144+00:00","updated_at":"2026-05-17T23:46:50.938144+00:00"}