{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:PAQNEBBWG6QYLPO5UBQFOHZJND","short_pith_number":"pith:PAQNEBBW","canonical_record":{"source":{"id":"0908.3349","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-08-24T01:01:58Z","cross_cats_sorted":[],"title_canon_sha256":"1e62804dd62502b171abbd2120d98e76fbac4f813ed98e8efa890b5c8b77ecd7","abstract_canon_sha256":"73f304d0886699ce23f9f4c869bf70d46d28ea09ea877a149c09485a7b0b8948"},"schema_version":"1.0"},"canonical_sha256":"7820d2043637a185bddda060571f2968c53b3e9de7edd2c77b90293a114d7b12","source":{"kind":"arxiv","id":"0908.3349","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0908.3349","created_at":"2026-05-18T02:12:23Z"},{"alias_kind":"arxiv_version","alias_value":"0908.3349v3","created_at":"2026-05-18T02:12:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.3349","created_at":"2026-05-18T02:12:23Z"},{"alias_kind":"pith_short_12","alias_value":"PAQNEBBWG6QY","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"PAQNEBBWG6QYLPO5","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"PAQNEBBW","created_at":"2026-05-18T12:26:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:PAQNEBBWG6QYLPO5UBQFOHZJND","target":"record","payload":{"canonical_record":{"source":{"id":"0908.3349","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-08-24T01:01:58Z","cross_cats_sorted":[],"title_canon_sha256":"1e62804dd62502b171abbd2120d98e76fbac4f813ed98e8efa890b5c8b77ecd7","abstract_canon_sha256":"73f304d0886699ce23f9f4c869bf70d46d28ea09ea877a149c09485a7b0b8948"},"schema_version":"1.0"},"canonical_sha256":"7820d2043637a185bddda060571f2968c53b3e9de7edd2c77b90293a114d7b12","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:12:23.765038Z","signature_b64":"syWb1mWrXl+3LaixZ6GA2wPVdMulrXEUGfxWyf9nM5xfePkHOr4++CoQCzw/woY00dS5Re9RSydlc9VuOv+NAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7820d2043637a185bddda060571f2968c53b3e9de7edd2c77b90293a114d7b12","last_reissued_at":"2026-05-18T02:12:23.764263Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:12:23.764263Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0908.3349","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:12:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kyUxSKdjCYYmpoCXhtXL0TeNyrovX+uZA+wNvuewRlAiL+QHP+qW53tj3UpAvxrX5LNYu/NRYz/YoQql3JIhAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T05:59:37.857499Z"},"content_sha256":"fa2ca8c8317b64db73e105f564badf5d1d19a0837ded3398f31e8366a5a1615f","schema_version":"1.0","event_id":"sha256:fa2ca8c8317b64db73e105f564badf5d1d19a0837ded3398f31e8366a5a1615f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:PAQNEBBWG6QYLPO5UBQFOHZJND","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An alternative approach to regularity for the Navier-Stokes equations in critical spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Carlos E. Kenig, Gabriel S. Koch","submitted_at":"2009-08-24T01:01:58Z","abstract_excerpt":"In this paper we present an alternative viewpoint on recent studies of regularity of solutions to the Navier-Stokes equations in critical spaces. In particular, we prove that mild solutions which remain bounded in the space $\\dot H^{1/2}$ do not become singular in finite time, a result which was proved in a more general setting by L. Escauriaza, G. Seregin and V. Sverak using a different approach. We use the method of \"concentration-compactness\" + \"rigidity theorem\" which was recently developed by C. Kenig and F. Merle to treat critical dispersive equations. To the authors' knowledge, this is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.3349","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:12:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e15TAmgnPfPw+laIx/RO4/HEJJwhr3ZdAB28zhMNfS6z6BBXsYP/sNa0POZWTjBbE9QhrtNDLgyEx+4yLloxBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T05:59:37.858159Z"},"content_sha256":"e8022250bfea42fb6c33943a8fbca065cd6da1958353cc0a16de757b3e76a9cd","schema_version":"1.0","event_id":"sha256:e8022250bfea42fb6c33943a8fbca065cd6da1958353cc0a16de757b3e76a9cd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PAQNEBBWG6QYLPO5UBQFOHZJND/bundle.json","state_url":"https://pith.science/pith/PAQNEBBWG6QYLPO5UBQFOHZJND/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PAQNEBBWG6QYLPO5UBQFOHZJND/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-05T05:59:37Z","links":{"resolver":"https://pith.science/pith/PAQNEBBWG6QYLPO5UBQFOHZJND","bundle":"https://pith.science/pith/PAQNEBBWG6QYLPO5UBQFOHZJND/bundle.json","state":"https://pith.science/pith/PAQNEBBWG6QYLPO5UBQFOHZJND/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PAQNEBBWG6QYLPO5UBQFOHZJND/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:PAQNEBBWG6QYLPO5UBQFOHZJND","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"73f304d0886699ce23f9f4c869bf70d46d28ea09ea877a149c09485a7b0b8948","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-08-24T01:01:58Z","title_canon_sha256":"1e62804dd62502b171abbd2120d98e76fbac4f813ed98e8efa890b5c8b77ecd7"},"schema_version":"1.0","source":{"id":"0908.3349","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0908.3349","created_at":"2026-05-18T02:12:23Z"},{"alias_kind":"arxiv_version","alias_value":"0908.3349v3","created_at":"2026-05-18T02:12:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.3349","created_at":"2026-05-18T02:12:23Z"},{"alias_kind":"pith_short_12","alias_value":"PAQNEBBWG6QY","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"PAQNEBBWG6QYLPO5","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"PAQNEBBW","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:e8022250bfea42fb6c33943a8fbca065cd6da1958353cc0a16de757b3e76a9cd","target":"graph","created_at":"2026-05-18T02:12:23Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper we present an alternative viewpoint on recent studies of regularity of solutions to the Navier-Stokes equations in critical spaces. In particular, we prove that mild solutions which remain bounded in the space $\\dot H^{1/2}$ do not become singular in finite time, a result which was proved in a more general setting by L. Escauriaza, G. Seregin and V. Sverak using a different approach. We use the method of \"concentration-compactness\" + \"rigidity theorem\" which was recently developed by C. Kenig and F. Merle to treat critical dispersive equations. To the authors' knowledge, this is ","authors_text":"Carlos E. Kenig, Gabriel S. Koch","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-08-24T01:01:58Z","title":"An alternative approach to regularity for the Navier-Stokes equations in critical spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.3349","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:fa2ca8c8317b64db73e105f564badf5d1d19a0837ded3398f31e8366a5a1615f","target":"record","created_at":"2026-05-18T02:12:23Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"73f304d0886699ce23f9f4c869bf70d46d28ea09ea877a149c09485a7b0b8948","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-08-24T01:01:58Z","title_canon_sha256":"1e62804dd62502b171abbd2120d98e76fbac4f813ed98e8efa890b5c8b77ecd7"},"schema_version":"1.0","source":{"id":"0908.3349","kind":"arxiv","version":3}},"canonical_sha256":"7820d2043637a185bddda060571f2968c53b3e9de7edd2c77b90293a114d7b12","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7820d2043637a185bddda060571f2968c53b3e9de7edd2c77b90293a114d7b12","first_computed_at":"2026-05-18T02:12:23.764263Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:12:23.764263Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"syWb1mWrXl+3LaixZ6GA2wPVdMulrXEUGfxWyf9nM5xfePkHOr4++CoQCzw/woY00dS5Re9RSydlc9VuOv+NAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:12:23.765038Z","signed_message":"canonical_sha256_bytes"},"source_id":"0908.3349","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fa2ca8c8317b64db73e105f564badf5d1d19a0837ded3398f31e8366a5a1615f","sha256:e8022250bfea42fb6c33943a8fbca065cd6da1958353cc0a16de757b3e76a9cd"],"state_sha256":"e8d53ab74786411d67849cefd439f9b5942e118fdcbf5cf218629b74b7a4d959"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6Gz0+NlCqUUbHHJOrRz1prxva/ByWR7pQg1N6qA/RQbMOZf+ldVis4932AgBcCCQmsZFuzICybvYhbKYWAo1CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T05:59:37.863725Z","bundle_sha256":"b3705bbb9164d24bc17b9d10a728ab0ba4ecc4a5383225ef049b986997a3df50"}}