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Previously, this uniqueness result was obtained by Rivi\\`{e}re (when $M$ is the round sphere and the energy of initial data is small) and Freire (when $M$ is an arbitrary closed Riemannian manifold), given that $u_0\\in H^1(B_1,M)$ and $\\gamma=u_0|_{\\partial B_1}\\in "},"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":"1010.3313","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-10-16T03:46:18Z","cross_cats_sorted":[],"title_canon_sha256":"c496cf38953cddfa74e390651984104e1816818eab1e6a139346db4b88ed47ad","abstract_canon_sha256":"21e34022984c913b8c7a3d9dfa0857ddeede20237c09125a24cf35a916c36cd1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:09.499330Z","signature_b64":"dmPmcHvOd5IOqH0LTksm2LxXvihoDU64NQejHmj1AHwrTsWsa9Fj9u7yrnsHDNLhYdOVSPiiWpjBYBLft80mCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c689117b5121fe1c5f369cc90a54821fd7122b56a15e79f4512b457f4d609cf3","last_reissued_at":"2026-05-18T04:39:09.498606Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:09.498606Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Harmonic map heat flow with rough boundary data","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Lu Wang","submitted_at":"2010-10-16T03:46:18Z","abstract_excerpt":"Let $B_1$ be the unit open disk in $\\Real^2$ and $M$ be a closed Riemannian manifold. In this note, we first prove the uniqueness for weak solutions of the harmonic map heat flow in $H^1([0,T]\\times B_1,M)$ whose energy is non-increasing in time, given initial data $u_0\\in H^1(B_1,M)$ and boundary data $\\gamma=u_0|_{\\partial B_1}$. Previously, this uniqueness result was obtained by Rivi\\`{e}re (when $M$ is the round sphere and the energy of initial data is small) and Freire (when $M$ is an arbitrary closed Riemannian manifold), given that $u_0\\in H^1(B_1,M)$ and $\\gamma=u_0|_{\\partial B_1}\\in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.3313","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":"1010.3313","created_at":"2026-05-18T04:39:09.498720+00:00"},{"alias_kind":"arxiv_version","alias_value":"1010.3313v1","created_at":"2026-05-18T04:39:09.498720+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.3313","created_at":"2026-05-18T04:39:09.498720+00:00"},{"alias_kind":"pith_short_12","alias_value":"Y2ERC62REH7B","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_16","alias_value":"Y2ERC62REH7BYXZW","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_8","alias_value":"Y2ERC62R","created_at":"2026-05-18T12:26:17.028572+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/Y2ERC62REH7BYXZWTTEQUVECD7","json":"https://pith.science/pith/Y2ERC62REH7BYXZWTTEQUVECD7.json","graph_json":"https://pith.science/api/pith-number/Y2ERC62REH7BYXZWTTEQUVECD7/graph.json","events_json":"https://pith.science/api/pith-number/Y2ERC62REH7BYXZWTTEQUVECD7/events.json","paper":"https://pith.science/paper/Y2ERC62R"},"agent_actions":{"view_html":"https://pith.science/pith/Y2ERC62REH7BYXZWTTEQUVECD7","download_json":"https://pith.science/pith/Y2ERC62REH7BYXZWTTEQUVECD7.json","view_paper":"https://pith.science/paper/Y2ERC62R","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1010.3313&json=true","fetch_graph":"https://pith.science/api/pith-number/Y2ERC62REH7BYXZWTTEQUVECD7/graph.json","fetch_events":"https://pith.science/api/pith-number/Y2ERC62REH7BYXZWTTEQUVECD7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Y2ERC62REH7BYXZWTTEQUVECD7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Y2ERC62REH7BYXZWTTEQUVECD7/action/storage_attestation","attest_author":"https://pith.science/pith/Y2ERC62REH7BYXZWTTEQUVECD7/action/author_attestation","sign_citation":"https://pith.science/pith/Y2ERC62REH7BYXZWTTEQUVECD7/action/citation_signature","submit_replication":"https://pith.science/pith/Y2ERC62REH7BYXZWTTEQUVECD7/action/replication_record"}},"created_at":"2026-05-18T04:39:09.498720+00:00","updated_at":"2026-05-18T04:39:09.498720+00:00"}