{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:GGZX2Y5H7MEDFK776MGXRVXXUS","short_pith_number":"pith:GGZX2Y5H","canonical_record":{"source":{"id":"1306.5342","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-06-22T18:14:32Z","cross_cats_sorted":[],"title_canon_sha256":"3b04f9e1c7d105729ba8df3dece1a6cdf38fdf61f68f3caa7fe7a8a1a3cbae5a","abstract_canon_sha256":"903c8db1a20264a50d3061ed46f5635aac84619103669956e42e3bfa630a16fa"},"schema_version":"1.0"},"canonical_sha256":"31b37d63a7fb0832abfff30d78d6f7a4a9ec586d7395bf59c76dd61366b642b4","source":{"kind":"arxiv","id":"1306.5342","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.5342","created_at":"2026-05-18T03:20:08Z"},{"alias_kind":"arxiv_version","alias_value":"1306.5342v1","created_at":"2026-05-18T03:20:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.5342","created_at":"2026-05-18T03:20:08Z"},{"alias_kind":"pith_short_12","alias_value":"GGZX2Y5H7MED","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"GGZX2Y5H7MEDFK77","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"GGZX2Y5H","created_at":"2026-05-18T12:27:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:GGZX2Y5H7MEDFK776MGXRVXXUS","target":"record","payload":{"canonical_record":{"source":{"id":"1306.5342","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-06-22T18:14:32Z","cross_cats_sorted":[],"title_canon_sha256":"3b04f9e1c7d105729ba8df3dece1a6cdf38fdf61f68f3caa7fe7a8a1a3cbae5a","abstract_canon_sha256":"903c8db1a20264a50d3061ed46f5635aac84619103669956e42e3bfa630a16fa"},"schema_version":"1.0"},"canonical_sha256":"31b37d63a7fb0832abfff30d78d6f7a4a9ec586d7395bf59c76dd61366b642b4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:20:08.476909Z","signature_b64":"ZZqkBIXLi3O6OxGlr9vZqTrMPGNhRk2iGJQEbMyw5MjnbVO/7pc2YEiqmFSurD7BIJIL7KH6nVLn5WrcKPbHAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"31b37d63a7fb0832abfff30d78d6f7a4a9ec586d7395bf59c76dd61366b642b4","last_reissued_at":"2026-05-18T03:20:08.476370Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:20:08.476370Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1306.5342","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":"2026-05-18T03:20:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jo5rVLCM45yZA/NW+Jq2osYHLsuukQviKR3wePx5X2JMNGV+8PgZdnoGsd/w8Rksu9TH3a+umF8RrRETi62wCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T23:03:40.298727Z"},"content_sha256":"0010ba48b769497084e504d55d314f08e4fdd245c2e349963ddb1ebe20babbcc","schema_version":"1.0","event_id":"sha256:0010ba48b769497084e504d55d314f08e4fdd245c2e349963ddb1ebe20babbcc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:GGZX2Y5H7MEDFK776MGXRVXXUS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stochastic hydrodynamic-type evolution equations driven by L\\'{e}vy noise in 3D unbounded domains - abstract framework and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"El\\.zbieta Motyl","submitted_at":"2013-06-22T18:14:32Z","abstract_excerpt":"The existence of martingale solutions of the hydrodynamic-type equations in 3D possibly unbounded domains is proved. The construction of the solution is based on the Faedo-Galerkin approximation. To overcome the difficulty related to the lack of the compactness of Sobolev embeddings in the case of unbounded domain we use certain Fr\\'{e}chet space.\n  We use also compactness and tightness criteria in some nonmetrizable spaces and a version of the Skorokhod Theorem in non-metric spaces. The general framework is applied to the stochastic Navier-Stokes, magneto-hydrodynamic (MHD) and the Boussinesq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.5342","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":"2026-05-18T03:20:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VtPTprkLZiEzg1bgZJ9vWuz1wLXKDpsbtPEXANA76xpm2Vi4D68p17tpqlBvMO6IAqSV8nBLmCWPozb24ixRAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T23:03:40.299415Z"},"content_sha256":"e4aba291c3d242be9fcf4fefa777e38065a14bb5422630ac631289c495c31d23","schema_version":"1.0","event_id":"sha256:e4aba291c3d242be9fcf4fefa777e38065a14bb5422630ac631289c495c31d23"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GGZX2Y5H7MEDFK776MGXRVXXUS/bundle.json","state_url":"https://pith.science/pith/GGZX2Y5H7MEDFK776MGXRVXXUS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GGZX2Y5H7MEDFK776MGXRVXXUS/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":"2026-06-10T23:03:40Z","links":{"resolver":"https://pith.science/pith/GGZX2Y5H7MEDFK776MGXRVXXUS","bundle":"https://pith.science/pith/GGZX2Y5H7MEDFK776MGXRVXXUS/bundle.json","state":"https://pith.science/pith/GGZX2Y5H7MEDFK776MGXRVXXUS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GGZX2Y5H7MEDFK776MGXRVXXUS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:GGZX2Y5H7MEDFK776MGXRVXXUS","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":"903c8db1a20264a50d3061ed46f5635aac84619103669956e42e3bfa630a16fa","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-06-22T18:14:32Z","title_canon_sha256":"3b04f9e1c7d105729ba8df3dece1a6cdf38fdf61f68f3caa7fe7a8a1a3cbae5a"},"schema_version":"1.0","source":{"id":"1306.5342","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.5342","created_at":"2026-05-18T03:20:08Z"},{"alias_kind":"arxiv_version","alias_value":"1306.5342v1","created_at":"2026-05-18T03:20:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.5342","created_at":"2026-05-18T03:20:08Z"},{"alias_kind":"pith_short_12","alias_value":"GGZX2Y5H7MED","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"GGZX2Y5H7MEDFK77","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"GGZX2Y5H","created_at":"2026-05-18T12:27:45Z"}],"graph_snapshots":[{"event_id":"sha256:e4aba291c3d242be9fcf4fefa777e38065a14bb5422630ac631289c495c31d23","target":"graph","created_at":"2026-05-18T03:20:08Z","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":"The existence of martingale solutions of the hydrodynamic-type equations in 3D possibly unbounded domains is proved. The construction of the solution is based on the Faedo-Galerkin approximation. To overcome the difficulty related to the lack of the compactness of Sobolev embeddings in the case of unbounded domain we use certain Fr\\'{e}chet space.\n  We use also compactness and tightness criteria in some nonmetrizable spaces and a version of the Skorokhod Theorem in non-metric spaces. The general framework is applied to the stochastic Navier-Stokes, magneto-hydrodynamic (MHD) and the Boussinesq","authors_text":"El\\.zbieta Motyl","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-06-22T18:14:32Z","title":"Stochastic hydrodynamic-type evolution equations driven by L\\'{e}vy noise in 3D unbounded domains - abstract framework and applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.5342","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":"sha256:0010ba48b769497084e504d55d314f08e4fdd245c2e349963ddb1ebe20babbcc","target":"record","created_at":"2026-05-18T03:20:08Z","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":"903c8db1a20264a50d3061ed46f5635aac84619103669956e42e3bfa630a16fa","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-06-22T18:14:32Z","title_canon_sha256":"3b04f9e1c7d105729ba8df3dece1a6cdf38fdf61f68f3caa7fe7a8a1a3cbae5a"},"schema_version":"1.0","source":{"id":"1306.5342","kind":"arxiv","version":1}},"canonical_sha256":"31b37d63a7fb0832abfff30d78d6f7a4a9ec586d7395bf59c76dd61366b642b4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"31b37d63a7fb0832abfff30d78d6f7a4a9ec586d7395bf59c76dd61366b642b4","first_computed_at":"2026-05-18T03:20:08.476370Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:20:08.476370Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZZqkBIXLi3O6OxGlr9vZqTrMPGNhRk2iGJQEbMyw5MjnbVO/7pc2YEiqmFSurD7BIJIL7KH6nVLn5WrcKPbHAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:20:08.476909Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.5342","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0010ba48b769497084e504d55d314f08e4fdd245c2e349963ddb1ebe20babbcc","sha256:e4aba291c3d242be9fcf4fefa777e38065a14bb5422630ac631289c495c31d23"],"state_sha256":"94df6309cbf4660c8780d0c6783ec62cbc58f95ba984430cab68fa493dc29fa1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XKJ/6P1qi1clWIIV6x0GkXYJhyMl9Iphu8NqOGKj918f52R/Y9EtZ1MqAAP5G+2hUMDQlp0HrkbnGwVuybP3CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T23:03:40.302918Z","bundle_sha256":"3ab5ef1260425db646985247466d5b33abc2c1ec4635f34d7fadca8aa050086a"}}