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Namely, either the system is locally ill-posed in $H^{m}(\\mathbb{R}^3)$, or it is locally well-posed, but there exists an initial data in $H^{m}(\\mathbb{R}^3)$, for which the $H^{m}(\\mathbb{R}^3)$ norm of solution blows-up in finite time if $m>7/2$.\n  In the latter case we choose an axisymmetric initial data $u_0(x)=u_{0r}(r,z)e_r+ b_{0z}(r,z)e_z$ and $B_0(x)=b_{0\\theta}(r,z)e_{\\theta}$, and reduce the system to"},"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":"1312.5519","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-12-19T12:51:06Z","cross_cats_sorted":[],"title_canon_sha256":"a6d81950dfa245605c941a5d575cd917a3a7dc03593a1a4f8f9de7db067cfe52","abstract_canon_sha256":"04830479bc44f90a5aa0d4a06e7c9d7105858c2b646f8729b48bf26caa61589f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:04:14.613652Z","signature_b64":"toXPQgG6JQfemS9nP1uuCVlrsZVBmisMW1NOa2JBNI10vBGMD7DXGG6XXu6mb1ZPy7JAvJGc/XuHB8ISwKBUAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"25aed4d69e95ec7412374834c7f1133c51faa469107e3a514e742cb7e32679b3","last_reissued_at":"2026-05-18T03:04:14.613152Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:04:14.613152Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Singularity formation for the incompressible Hall-MHD equations without resistivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dongho Chae, Shangkun Weng","submitted_at":"2013-12-19T12:51:06Z","abstract_excerpt":"In this paper we show that the incompressible Hall-MHD system without resistivity is not globally in time well-posed in any Sobolev space $H^{m}(\\mathbb{R}^3)$ for any $m>\\frac{7}{2}$. Namely, either the system is locally ill-posed in $H^{m}(\\mathbb{R}^3)$, or it is locally well-posed, but there exists an initial data in $H^{m}(\\mathbb{R}^3)$, for which the $H^{m}(\\mathbb{R}^3)$ norm of solution blows-up in finite time if $m>7/2$.\n  In the latter case we choose an axisymmetric initial data $u_0(x)=u_{0r}(r,z)e_r+ b_{0z}(r,z)e_z$ and $B_0(x)=b_{0\\theta}(r,z)e_{\\theta}$, and reduce the system to"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.5519","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":"1312.5519","created_at":"2026-05-18T03:04:14.613242+00:00"},{"alias_kind":"arxiv_version","alias_value":"1312.5519v1","created_at":"2026-05-18T03:04:14.613242+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.5519","created_at":"2026-05-18T03:04:14.613242+00:00"},{"alias_kind":"pith_short_12","alias_value":"EWXNJVU6SXWH","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_16","alias_value":"EWXNJVU6SXWHIERX","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_8","alias_value":"EWXNJVU6","created_at":"2026-05-18T12:27:43.054852+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/EWXNJVU6SXWHIERXJA2MP4ITHR","json":"https://pith.science/pith/EWXNJVU6SXWHIERXJA2MP4ITHR.json","graph_json":"https://pith.science/api/pith-number/EWXNJVU6SXWHIERXJA2MP4ITHR/graph.json","events_json":"https://pith.science/api/pith-number/EWXNJVU6SXWHIERXJA2MP4ITHR/events.json","paper":"https://pith.science/paper/EWXNJVU6"},"agent_actions":{"view_html":"https://pith.science/pith/EWXNJVU6SXWHIERXJA2MP4ITHR","download_json":"https://pith.science/pith/EWXNJVU6SXWHIERXJA2MP4ITHR.json","view_paper":"https://pith.science/paper/EWXNJVU6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1312.5519&json=true","fetch_graph":"https://pith.science/api/pith-number/EWXNJVU6SXWHIERXJA2MP4ITHR/graph.json","fetch_events":"https://pith.science/api/pith-number/EWXNJVU6SXWHIERXJA2MP4ITHR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EWXNJVU6SXWHIERXJA2MP4ITHR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EWXNJVU6SXWHIERXJA2MP4ITHR/action/storage_attestation","attest_author":"https://pith.science/pith/EWXNJVU6SXWHIERXJA2MP4ITHR/action/author_attestation","sign_citation":"https://pith.science/pith/EWXNJVU6SXWHIERXJA2MP4ITHR/action/citation_signature","submit_replication":"https://pith.science/pith/EWXNJVU6SXWHIERXJA2MP4ITHR/action/replication_record"}},"created_at":"2026-05-18T03:04:14.613242+00:00","updated_at":"2026-05-18T03:04:14.613242+00:00"}