{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:FOTL32NOSYKNKOEYASLPJPU3LX","short_pith_number":"pith:FOTL32NO","canonical_record":{"source":{"id":"1411.5156","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-11-19T09:24:00Z","cross_cats_sorted":[],"title_canon_sha256":"52dfb467421b4208c97a0a2207c32f0fbe11944cabe860d5d0c98dffc8cf7458","abstract_canon_sha256":"8e51e42a8e6467e9d9f325a3d340a74b75000c78ae93707232622097aa43964a"},"schema_version":"1.0"},"canonical_sha256":"2ba6bde9ae9614d538980496f4be9b5dea0f92ec1ff07aaa49b88eec19c1c665","source":{"kind":"arxiv","id":"1411.5156","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.5156","created_at":"2026-05-18T02:34:44Z"},{"alias_kind":"arxiv_version","alias_value":"1411.5156v1","created_at":"2026-05-18T02:34:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.5156","created_at":"2026-05-18T02:34:44Z"},{"alias_kind":"pith_short_12","alias_value":"FOTL32NOSYKN","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"FOTL32NOSYKNKOEY","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"FOTL32NO","created_at":"2026-05-18T12:28:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:FOTL32NOSYKNKOEYASLPJPU3LX","target":"record","payload":{"canonical_record":{"source":{"id":"1411.5156","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-11-19T09:24:00Z","cross_cats_sorted":[],"title_canon_sha256":"52dfb467421b4208c97a0a2207c32f0fbe11944cabe860d5d0c98dffc8cf7458","abstract_canon_sha256":"8e51e42a8e6467e9d9f325a3d340a74b75000c78ae93707232622097aa43964a"},"schema_version":"1.0"},"canonical_sha256":"2ba6bde9ae9614d538980496f4be9b5dea0f92ec1ff07aaa49b88eec19c1c665","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:34:44.266401Z","signature_b64":"J5SJo2Gn4Tyorl+yJ5j8cYHunB9RqLHH/YF0tkZnXGlYl5h+sH3zcDQPQSnzkkHq7e/hjoy4kWWUICv9BldTCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2ba6bde9ae9614d538980496f4be9b5dea0f92ec1ff07aaa49b88eec19c1c665","last_reissued_at":"2026-05-18T02:34:44.265734Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:34:44.265734Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1411.5156","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:34:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QraWoQFPZMrmo5Zc05jqeMRqyBCADTCU6wAOVCgnCLJ5M8w8/2lqbdec0Qguz9/fOCQ856PTu6ZeoERdbGCIBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T08:33:19.977044Z"},"content_sha256":"8db14abdf1ccf0a5157f308ddad9af75540f15886e0d1184e75c231fa8b9a2b7","schema_version":"1.0","event_id":"sha256:8db14abdf1ccf0a5157f308ddad9af75540f15886e0d1184e75c231fa8b9a2b7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:FOTL32NOSYKNKOEYASLPJPU3LX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Infinite energy solutions of the two-dimensional Navier-Stokes equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Thierry Gallay","submitted_at":"2014-11-19T09:24:00Z","abstract_excerpt":"These notes are based on a series of lectures delivered by the author at the University of Toulouse in February 2014. They are entirely devoted to the initial value problem and the long-time behavior of solutions for the two-dimensional incompressible Navier-Stokes equations, in the particular case where the domain occupied by the fluid is the whole plane $\\mathbb{R}^2$ and the velocity field is only assumed to be bounded. In this context, local well-posedness is not difficult to establish, and a priori estimates on the vorticity distribution imply that all solutions are global and grow at mos"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.5156","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:34:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C5+zhVl2YXQRF32ta4YDVsOlyARwg1ItMgsd2I1V3Uw+jI6uUZ92g80KpE+wpFNzek9dBwjI9SYTWAaNuhfQDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T08:33:19.977403Z"},"content_sha256":"5ae83c9cd9e66ed82b6296881e1abebcb919f98ec3692518c5da4197baeb88b5","schema_version":"1.0","event_id":"sha256:5ae83c9cd9e66ed82b6296881e1abebcb919f98ec3692518c5da4197baeb88b5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FOTL32NOSYKNKOEYASLPJPU3LX/bundle.json","state_url":"https://pith.science/pith/FOTL32NOSYKNKOEYASLPJPU3LX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FOTL32NOSYKNKOEYASLPJPU3LX/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":"2026-05-28T08:33:19Z","links":{"resolver":"https://pith.science/pith/FOTL32NOSYKNKOEYASLPJPU3LX","bundle":"https://pith.science/pith/FOTL32NOSYKNKOEYASLPJPU3LX/bundle.json","state":"https://pith.science/pith/FOTL32NOSYKNKOEYASLPJPU3LX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FOTL32NOSYKNKOEYASLPJPU3LX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:FOTL32NOSYKNKOEYASLPJPU3LX","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":"8e51e42a8e6467e9d9f325a3d340a74b75000c78ae93707232622097aa43964a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-11-19T09:24:00Z","title_canon_sha256":"52dfb467421b4208c97a0a2207c32f0fbe11944cabe860d5d0c98dffc8cf7458"},"schema_version":"1.0","source":{"id":"1411.5156","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.5156","created_at":"2026-05-18T02:34:44Z"},{"alias_kind":"arxiv_version","alias_value":"1411.5156v1","created_at":"2026-05-18T02:34:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.5156","created_at":"2026-05-18T02:34:44Z"},{"alias_kind":"pith_short_12","alias_value":"FOTL32NOSYKN","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"FOTL32NOSYKNKOEY","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"FOTL32NO","created_at":"2026-05-18T12:28:28Z"}],"graph_snapshots":[{"event_id":"sha256:5ae83c9cd9e66ed82b6296881e1abebcb919f98ec3692518c5da4197baeb88b5","target":"graph","created_at":"2026-05-18T02:34:44Z","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":"These notes are based on a series of lectures delivered by the author at the University of Toulouse in February 2014. They are entirely devoted to the initial value problem and the long-time behavior of solutions for the two-dimensional incompressible Navier-Stokes equations, in the particular case where the domain occupied by the fluid is the whole plane $\\mathbb{R}^2$ and the velocity field is only assumed to be bounded. In this context, local well-posedness is not difficult to establish, and a priori estimates on the vorticity distribution imply that all solutions are global and grow at mos","authors_text":"Thierry Gallay","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-11-19T09:24:00Z","title":"Infinite energy solutions of the two-dimensional Navier-Stokes equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.5156","kind":"arxiv","version":1},"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":"sha256:8db14abdf1ccf0a5157f308ddad9af75540f15886e0d1184e75c231fa8b9a2b7","target":"record","created_at":"2026-05-18T02:34:44Z","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":"8e51e42a8e6467e9d9f325a3d340a74b75000c78ae93707232622097aa43964a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-11-19T09:24:00Z","title_canon_sha256":"52dfb467421b4208c97a0a2207c32f0fbe11944cabe860d5d0c98dffc8cf7458"},"schema_version":"1.0","source":{"id":"1411.5156","kind":"arxiv","version":1}},"canonical_sha256":"2ba6bde9ae9614d538980496f4be9b5dea0f92ec1ff07aaa49b88eec19c1c665","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2ba6bde9ae9614d538980496f4be9b5dea0f92ec1ff07aaa49b88eec19c1c665","first_computed_at":"2026-05-18T02:34:44.265734Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:34:44.265734Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"J5SJo2Gn4Tyorl+yJ5j8cYHunB9RqLHH/YF0tkZnXGlYl5h+sH3zcDQPQSnzkkHq7e/hjoy4kWWUICv9BldTCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:34:44.266401Z","signed_message":"canonical_sha256_bytes"},"source_id":"1411.5156","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8db14abdf1ccf0a5157f308ddad9af75540f15886e0d1184e75c231fa8b9a2b7","sha256:5ae83c9cd9e66ed82b6296881e1abebcb919f98ec3692518c5da4197baeb88b5"],"state_sha256":"980cf4299de6faef59cb308f9d51c7b34d9cb2e941e69986466a4138bd224a3c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cX4oIQpNyfvhmQW0/n4y2XA1S5cqsa88PXvgquZrTy8H20Z0lNVDl3lSo4I2+Hons5CwsqOTxO86Pyv20CsFBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T08:33:19.979354Z","bundle_sha256":"adee2524945dafdf6476f83a3547ebfc023334e5b5f2e88576fd8718f71d3bdd"}}