{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:XQ2H2YQIVXLIVKPWACPLRC6JVO","short_pith_number":"pith:XQ2H2YQI","canonical_record":{"source":{"id":"1103.0796","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-03-03T22:53:55Z","cross_cats_sorted":[],"title_canon_sha256":"17667d551fd476991e754030892357884f25c9f658331e9fe6fc4ba03b35c5e5","abstract_canon_sha256":"136f798a5bbbf00a93325bcc03dde61f39368bc3dd5c8b529830900b496f6a82"},"schema_version":"1.0"},"canonical_sha256":"bc347d6208add68aa9f6009eb88bc9aba935328724a8634f505b3ec809d64c37","source":{"kind":"arxiv","id":"1103.0796","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.0796","created_at":"2026-05-18T04:27:26Z"},{"alias_kind":"arxiv_version","alias_value":"1103.0796v1","created_at":"2026-05-18T04:27:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.0796","created_at":"2026-05-18T04:27:26Z"},{"alias_kind":"pith_short_12","alias_value":"XQ2H2YQIVXLI","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"XQ2H2YQIVXLIVKPW","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"XQ2H2YQI","created_at":"2026-05-18T12:26:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:XQ2H2YQIVXLIVKPWACPLRC6JVO","target":"record","payload":{"canonical_record":{"source":{"id":"1103.0796","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-03-03T22:53:55Z","cross_cats_sorted":[],"title_canon_sha256":"17667d551fd476991e754030892357884f25c9f658331e9fe6fc4ba03b35c5e5","abstract_canon_sha256":"136f798a5bbbf00a93325bcc03dde61f39368bc3dd5c8b529830900b496f6a82"},"schema_version":"1.0"},"canonical_sha256":"bc347d6208add68aa9f6009eb88bc9aba935328724a8634f505b3ec809d64c37","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:27:26.253162Z","signature_b64":"UDKyD74bysGtvsjzLGANAhHhg2duG/ebi9nzOTl6eJCQGW9AilHbIQcNek9wNL6URdkueeAeNOMPQtXV35ZXCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bc347d6208add68aa9f6009eb88bc9aba935328724a8634f505b3ec809d64c37","last_reissued_at":"2026-05-18T04:27:26.252510Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:27:26.252510Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1103.0796","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":"2026-05-18T04:27:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LrkryNszRNI68K/kZM4Fu6jdwQrdvZ0FEIa/X1LPWpasurgPbLvx/dqhfWaIFQ07RtaIztTOL5de5oGM8xPQCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T09:20:08.979964Z"},"content_sha256":"24c2321328ebdac87eaac24eab6afd350e81e6f8ce6f05cdbd2dfbe7f17751d7","schema_version":"1.0","event_id":"sha256:24c2321328ebdac87eaac24eab6afd350e81e6f8ce6f05cdbd2dfbe7f17751d7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:XQ2H2YQIVXLIVKPWACPLRC6JVO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Singularity \\& Regularity Issues for Simplified Models of Turbulence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hani Ali, Zied Ammari","submitted_at":"2011-03-03T22:53:55Z","abstract_excerpt":"We consider a family of Leray-$\\alpha$ models with periodic boundary conditions in three space dimensions. Such models are a regularization, with respect to a parameter $\\theta$, of the Navier-Stokes equations. In particular, they share with the original equation (NS) the property of existence of global weak solutions. We establish an upper bound on the Hausdorff dimension of the time singular set of those weak solutions when $\\theta$ is subcritical. The result is an interpolation between the bound proved by Scheffer for the Navier-Stokes equations and the regularity result proved in \\cite{A01"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.0796","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":"2026-05-18T04:27:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qSCXynzULGG3eInVrBX4a8esQkNvQJqLobyeEkk7ladDqzqBZ63FL59NbJHncvRbpIebBy1tKG8WtMgUUGqeBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T09:20:08.980309Z"},"content_sha256":"700447d349493e87adcf0930f439b993712dd0340169e27e4d7b5acd9848d54e","schema_version":"1.0","event_id":"sha256:700447d349493e87adcf0930f439b993712dd0340169e27e4d7b5acd9848d54e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XQ2H2YQIVXLIVKPWACPLRC6JVO/bundle.json","state_url":"https://pith.science/pith/XQ2H2YQIVXLIVKPWACPLRC6JVO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XQ2H2YQIVXLIVKPWACPLRC6JVO/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":"2026-06-22T09:20:08Z","links":{"resolver":"https://pith.science/pith/XQ2H2YQIVXLIVKPWACPLRC6JVO","bundle":"https://pith.science/pith/XQ2H2YQIVXLIVKPWACPLRC6JVO/bundle.json","state":"https://pith.science/pith/XQ2H2YQIVXLIVKPWACPLRC6JVO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XQ2H2YQIVXLIVKPWACPLRC6JVO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:XQ2H2YQIVXLIVKPWACPLRC6JVO","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":"136f798a5bbbf00a93325bcc03dde61f39368bc3dd5c8b529830900b496f6a82","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-03-03T22:53:55Z","title_canon_sha256":"17667d551fd476991e754030892357884f25c9f658331e9fe6fc4ba03b35c5e5"},"schema_version":"1.0","source":{"id":"1103.0796","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.0796","created_at":"2026-05-18T04:27:26Z"},{"alias_kind":"arxiv_version","alias_value":"1103.0796v1","created_at":"2026-05-18T04:27:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.0796","created_at":"2026-05-18T04:27:26Z"},{"alias_kind":"pith_short_12","alias_value":"XQ2H2YQIVXLI","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"XQ2H2YQIVXLIVKPW","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"XQ2H2YQI","created_at":"2026-05-18T12:26:47Z"}],"graph_snapshots":[{"event_id":"sha256:700447d349493e87adcf0930f439b993712dd0340169e27e4d7b5acd9848d54e","target":"graph","created_at":"2026-05-18T04:27:26Z","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 consider a family of Leray-$\\alpha$ models with periodic boundary conditions in three space dimensions. Such models are a regularization, with respect to a parameter $\\theta$, of the Navier-Stokes equations. In particular, they share with the original equation (NS) the property of existence of global weak solutions. We establish an upper bound on the Hausdorff dimension of the time singular set of those weak solutions when $\\theta$ is subcritical. The result is an interpolation between the bound proved by Scheffer for the Navier-Stokes equations and the regularity result proved in \\cite{A01","authors_text":"Hani Ali, Zied Ammari","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-03-03T22:53:55Z","title":"Singularity \\& Regularity Issues for Simplified Models of Turbulence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.0796","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":"sha256:24c2321328ebdac87eaac24eab6afd350e81e6f8ce6f05cdbd2dfbe7f17751d7","target":"record","created_at":"2026-05-18T04:27:26Z","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":"136f798a5bbbf00a93325bcc03dde61f39368bc3dd5c8b529830900b496f6a82","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-03-03T22:53:55Z","title_canon_sha256":"17667d551fd476991e754030892357884f25c9f658331e9fe6fc4ba03b35c5e5"},"schema_version":"1.0","source":{"id":"1103.0796","kind":"arxiv","version":1}},"canonical_sha256":"bc347d6208add68aa9f6009eb88bc9aba935328724a8634f505b3ec809d64c37","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bc347d6208add68aa9f6009eb88bc9aba935328724a8634f505b3ec809d64c37","first_computed_at":"2026-05-18T04:27:26.252510Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:27:26.252510Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UDKyD74bysGtvsjzLGANAhHhg2duG/ebi9nzOTl6eJCQGW9AilHbIQcNek9wNL6URdkueeAeNOMPQtXV35ZXCw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:27:26.253162Z","signed_message":"canonical_sha256_bytes"},"source_id":"1103.0796","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:24c2321328ebdac87eaac24eab6afd350e81e6f8ce6f05cdbd2dfbe7f17751d7","sha256:700447d349493e87adcf0930f439b993712dd0340169e27e4d7b5acd9848d54e"],"state_sha256":"3abd9d45a70f285bbf67f4226aa6d944a4f81ad47965a7b18ad758f5a7dd90d2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9Q3n7cKCzOwJ4mrIxZ0SErJ0LWlLeuhVm9CVGQQCGwccN3GkFnOgKxmfH0TVfqJxXWJ49U8i2Vo8GQTzY0cRDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T09:20:08.982234Z","bundle_sha256":"0d939e82483e4706d2971e32f3aafdc021e2e6ef011904d4eb149799e9e48d42"}}