{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:RSHV5K73X75QN4WYOUGJOFYZNV","short_pith_number":"pith:RSHV5K73","canonical_record":{"source":{"id":"1806.01498","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-06-05T05:16:24Z","cross_cats_sorted":[],"title_canon_sha256":"e12499db21e36b428e28d3f45698e7622a4f609e21bc8d759b475cf059bc8d5d","abstract_canon_sha256":"3369c2f52e5c1af2e6a0cb3a1c251081b4af97e668152f460db99582d4af252c"},"schema_version":"1.0"},"canonical_sha256":"8c8f5eabfbbffb06f2d8750c9717196d7a11386b3f2205ae904661b133ef0c02","source":{"kind":"arxiv","id":"1806.01498","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.01498","created_at":"2026-05-18T00:14:12Z"},{"alias_kind":"arxiv_version","alias_value":"1806.01498v1","created_at":"2026-05-18T00:14:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.01498","created_at":"2026-05-18T00:14:12Z"},{"alias_kind":"pith_short_12","alias_value":"RSHV5K73X75Q","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"RSHV5K73X75QN4WY","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"RSHV5K73","created_at":"2026-05-18T12:32:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:RSHV5K73X75QN4WYOUGJOFYZNV","target":"record","payload":{"canonical_record":{"source":{"id":"1806.01498","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-06-05T05:16:24Z","cross_cats_sorted":[],"title_canon_sha256":"e12499db21e36b428e28d3f45698e7622a4f609e21bc8d759b475cf059bc8d5d","abstract_canon_sha256":"3369c2f52e5c1af2e6a0cb3a1c251081b4af97e668152f460db99582d4af252c"},"schema_version":"1.0"},"canonical_sha256":"8c8f5eabfbbffb06f2d8750c9717196d7a11386b3f2205ae904661b133ef0c02","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:12.774388Z","signature_b64":"hO3QtMeDP0e3iqdEqeUGiCbMa4vt54cKaI8LVGWNRR/aMcnbvT4tkeiiK0AKLhIP+ztciRse9DCX8Wkz5pMIAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8c8f5eabfbbffb06f2d8750c9717196d7a11386b3f2205ae904661b133ef0c02","last_reissued_at":"2026-05-18T00:14:12.773899Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:12.773899Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1806.01498","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":"2026-05-18T00:14:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pcr/rvLqQPAtYQ/L5RGQEwYgSa+xFVbO3ZUc3ICznAM5ej/6G4QRTZUUyxzt1Ma7DZIC03O9zxSAwmEU5eb2Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T06:14:57.957257Z"},"content_sha256":"eb907062d307167866eabe1ad9b3f4295934abc207139741a6f2043055d37236","schema_version":"1.0","event_id":"sha256:eb907062d307167866eabe1ad9b3f4295934abc207139741a6f2043055d37236"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:RSHV5K73X75QN4WYOUGJOFYZNV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Galerkin approximation and strong norm bounds for the stochastic Navier-Stokes equations with multiplicative noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Igor Kukavica, Kerem Ugurlu, Mohammed Ziane","submitted_at":"2018-06-05T05:16:24Z","abstract_excerpt":"We investigate the convergence of the Galerkin approximation for the stochastic Navier-Stokes equations in an open bounded domain $\\mathcal{O}$ with the non-slip boundary condition. We prove that\n  \\begin{equation*}\n  \\mathbb{E} \\left[ \\sup_{t \\in [0,T]} \\phi_1(\\lVert (u(t)-u^n(t))\n  \\rVert^2_V) \\right] \\rightarrow 0\n  \\end{equation*} as $n \\rightarrow \\infty$ for any deterministic time $T > 0$ and for a specified moment function $\\phi_1(x)$ where $u^n(t,x)$ denotes the Galerkin approximation of the solution $u(t,x)$. Also, we provide a result on uniform boundedness of the moment $\\mathbb{E} ["},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.01498","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":"2026-05-18T00:14:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HCA0aoppuZfVuqGLEMrsmB8zQkYY0zQyRG4ea+szj9BlsZ7yLh2dUtAebbzj5JvTM1csXr64Zw3yr2Ltgo2rDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T06:14:57.957653Z"},"content_sha256":"9e5a8188d47049cb14dd8fc92910d43ee5e8f94646400ce87d7ee88614d866b5","schema_version":"1.0","event_id":"sha256:9e5a8188d47049cb14dd8fc92910d43ee5e8f94646400ce87d7ee88614d866b5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RSHV5K73X75QN4WYOUGJOFYZNV/bundle.json","state_url":"https://pith.science/pith/RSHV5K73X75QN4WYOUGJOFYZNV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RSHV5K73X75QN4WYOUGJOFYZNV/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":"2026-05-27T06:14:57Z","links":{"resolver":"https://pith.science/pith/RSHV5K73X75QN4WYOUGJOFYZNV","bundle":"https://pith.science/pith/RSHV5K73X75QN4WYOUGJOFYZNV/bundle.json","state":"https://pith.science/pith/RSHV5K73X75QN4WYOUGJOFYZNV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RSHV5K73X75QN4WYOUGJOFYZNV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:RSHV5K73X75QN4WYOUGJOFYZNV","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":"3369c2f52e5c1af2e6a0cb3a1c251081b4af97e668152f460db99582d4af252c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-06-05T05:16:24Z","title_canon_sha256":"e12499db21e36b428e28d3f45698e7622a4f609e21bc8d759b475cf059bc8d5d"},"schema_version":"1.0","source":{"id":"1806.01498","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.01498","created_at":"2026-05-18T00:14:12Z"},{"alias_kind":"arxiv_version","alias_value":"1806.01498v1","created_at":"2026-05-18T00:14:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.01498","created_at":"2026-05-18T00:14:12Z"},{"alias_kind":"pith_short_12","alias_value":"RSHV5K73X75Q","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"RSHV5K73X75QN4WY","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"RSHV5K73","created_at":"2026-05-18T12:32:50Z"}],"graph_snapshots":[{"event_id":"sha256:9e5a8188d47049cb14dd8fc92910d43ee5e8f94646400ce87d7ee88614d866b5","target":"graph","created_at":"2026-05-18T00:14:12Z","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":"We investigate the convergence of the Galerkin approximation for the stochastic Navier-Stokes equations in an open bounded domain $\\mathcal{O}$ with the non-slip boundary condition. We prove that\n  \\begin{equation*}\n  \\mathbb{E} \\left[ \\sup_{t \\in [0,T]} \\phi_1(\\lVert (u(t)-u^n(t))\n  \\rVert^2_V) \\right] \\rightarrow 0\n  \\end{equation*} as $n \\rightarrow \\infty$ for any deterministic time $T > 0$ and for a specified moment function $\\phi_1(x)$ where $u^n(t,x)$ denotes the Galerkin approximation of the solution $u(t,x)$. Also, we provide a result on uniform boundedness of the moment $\\mathbb{E} [","authors_text":"Igor Kukavica, Kerem Ugurlu, Mohammed Ziane","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-06-05T05:16:24Z","title":"On the Galerkin approximation and strong norm bounds for the stochastic Navier-Stokes equations with multiplicative noise"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.01498","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:eb907062d307167866eabe1ad9b3f4295934abc207139741a6f2043055d37236","target":"record","created_at":"2026-05-18T00:14:12Z","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":"3369c2f52e5c1af2e6a0cb3a1c251081b4af97e668152f460db99582d4af252c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-06-05T05:16:24Z","title_canon_sha256":"e12499db21e36b428e28d3f45698e7622a4f609e21bc8d759b475cf059bc8d5d"},"schema_version":"1.0","source":{"id":"1806.01498","kind":"arxiv","version":1}},"canonical_sha256":"8c8f5eabfbbffb06f2d8750c9717196d7a11386b3f2205ae904661b133ef0c02","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8c8f5eabfbbffb06f2d8750c9717196d7a11386b3f2205ae904661b133ef0c02","first_computed_at":"2026-05-18T00:14:12.773899Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:12.773899Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hO3QtMeDP0e3iqdEqeUGiCbMa4vt54cKaI8LVGWNRR/aMcnbvT4tkeiiK0AKLhIP+ztciRse9DCX8Wkz5pMIAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:12.774388Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.01498","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:eb907062d307167866eabe1ad9b3f4295934abc207139741a6f2043055d37236","sha256:9e5a8188d47049cb14dd8fc92910d43ee5e8f94646400ce87d7ee88614d866b5"],"state_sha256":"7bc164195efe7671b2e730920c9d53758536609533e820891600881567c7a424"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CCyTJ+h/cCKf4edI1BQpaWYqZkyu6u3poxiSylksmTXk7INPoZDC46Uqi8pAQGs2uivurIaykAQp4BZodzamDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T06:14:57.960277Z","bundle_sha256":"51ac6f68e23df74bf3bed16c2282664cbb9f713e3af528a5f8fb4e3658beb72c"}}