{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:EUOKJXF2D7ZBTB35NZOBWE3Y7H","short_pith_number":"pith:EUOKJXF2","canonical_record":{"source":{"id":"2502.10024","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-02-14T09:08:23Z","cross_cats_sorted":[],"title_canon_sha256":"536fefee2156813a1a7d59d4b55578d170ab0b5570519181d4fd1762545cef33","abstract_canon_sha256":"982b4792d9dbb46899ecd4777c3424bb48916a4bb80067185bf2519f815a988e"},"schema_version":"1.0"},"canonical_sha256":"251ca4dcba1ff219877d6e5c1b1378f9caf72070b36f87f31926f569c9ea06e7","source":{"kind":"arxiv","id":"2502.10024","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.10024","created_at":"2026-07-05T10:14:16Z"},{"alias_kind":"arxiv_version","alias_value":"2502.10024v1","created_at":"2026-07-05T10:14:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.10024","created_at":"2026-07-05T10:14:16Z"},{"alias_kind":"pith_short_12","alias_value":"EUOKJXF2D7ZB","created_at":"2026-07-05T10:14:16Z"},{"alias_kind":"pith_short_16","alias_value":"EUOKJXF2D7ZBTB35","created_at":"2026-07-05T10:14:16Z"},{"alias_kind":"pith_short_8","alias_value":"EUOKJXF2","created_at":"2026-07-05T10:14:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:EUOKJXF2D7ZBTB35NZOBWE3Y7H","target":"record","payload":{"canonical_record":{"source":{"id":"2502.10024","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-02-14T09:08:23Z","cross_cats_sorted":[],"title_canon_sha256":"536fefee2156813a1a7d59d4b55578d170ab0b5570519181d4fd1762545cef33","abstract_canon_sha256":"982b4792d9dbb46899ecd4777c3424bb48916a4bb80067185bf2519f815a988e"},"schema_version":"1.0"},"canonical_sha256":"251ca4dcba1ff219877d6e5c1b1378f9caf72070b36f87f31926f569c9ea06e7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:14:16.240738Z","signature_b64":"8K6DHLx2w1xSUx7G2SiZJLadc5ZzPgFddjZcnQltTVlsJjK0z20nINoajKSf+z3uOE2Mcl3MRVKGqHepdGlAAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"251ca4dcba1ff219877d6e5c1b1378f9caf72070b36f87f31926f569c9ea06e7","last_reissued_at":"2026-07-05T10:14:16.240244Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:14:16.240244Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2502.10024","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":"2026-07-05T10:14:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v1lxZsdL+zGp+a0ELq2bXa8vk1yY1hi1yo4YonVTAnfKWsDMDt92wGIVADM3XQguxKpJEe2m3prudNUUCXDpDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T03:56:06.855673Z"},"content_sha256":"423b7d3abd7675a233a07a3f4f26cb5f27824dc53217d294af0126acea714654","schema_version":"1.0","event_id":"sha256:423b7d3abd7675a233a07a3f4f26cb5f27824dc53217d294af0126acea714654"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:EUOKJXF2D7ZBTB35NZOBWE3Y7H","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Geometric blow-up criteria for the non-homogeneous incompressible Euler equations in 2-D","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Francesco Fanelli","submitted_at":"2025-02-14T09:08:23Z","abstract_excerpt":"This paper concerns the study of the incompressible Euler equations with variable density, in the case of space dimension $d=2$. Contrarily to their homogeneous (constant density) counterpart, those equations are not known to be well-posed globally in time. A classical blow-up/continuation criterion for smooth solutions relies on the control of the Lipschitz norm of the velocity field $u$. Here we show that, for establishing blow-up or continuation of solutions, it is enough to determine a control of $\\nabla u$ only along the direction $X=\\nabla^\\perp\\rho$, where $\\rho$ represents the density "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.10024","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2502.10024/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T10:14:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yuiGR+DNGwiUfRw/IULVASD1eC8e4PRP88dJvc/JxYFpXYWtscrTe7kZrn228D11q7aEGptxdAxhQetAmmQ5Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T03:56:06.856191Z"},"content_sha256":"47746300fedbf9bb2035eeb5ae55e0dc86c0e9e14ab09af8df1c43bd8ac12a22","schema_version":"1.0","event_id":"sha256:47746300fedbf9bb2035eeb5ae55e0dc86c0e9e14ab09af8df1c43bd8ac12a22"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EUOKJXF2D7ZBTB35NZOBWE3Y7H/bundle.json","state_url":"https://pith.science/pith/EUOKJXF2D7ZBTB35NZOBWE3Y7H/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EUOKJXF2D7ZBTB35NZOBWE3Y7H/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":"2026-08-15T03:56:06Z","links":{"resolver":"https://pith.science/pith/EUOKJXF2D7ZBTB35NZOBWE3Y7H","bundle":"https://pith.science/pith/EUOKJXF2D7ZBTB35NZOBWE3Y7H/bundle.json","state":"https://pith.science/pith/EUOKJXF2D7ZBTB35NZOBWE3Y7H/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EUOKJXF2D7ZBTB35NZOBWE3Y7H/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:EUOKJXF2D7ZBTB35NZOBWE3Y7H","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":"982b4792d9dbb46899ecd4777c3424bb48916a4bb80067185bf2519f815a988e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-02-14T09:08:23Z","title_canon_sha256":"536fefee2156813a1a7d59d4b55578d170ab0b5570519181d4fd1762545cef33"},"schema_version":"1.0","source":{"id":"2502.10024","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.10024","created_at":"2026-07-05T10:14:16Z"},{"alias_kind":"arxiv_version","alias_value":"2502.10024v1","created_at":"2026-07-05T10:14:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.10024","created_at":"2026-07-05T10:14:16Z"},{"alias_kind":"pith_short_12","alias_value":"EUOKJXF2D7ZB","created_at":"2026-07-05T10:14:16Z"},{"alias_kind":"pith_short_16","alias_value":"EUOKJXF2D7ZBTB35","created_at":"2026-07-05T10:14:16Z"},{"alias_kind":"pith_short_8","alias_value":"EUOKJXF2","created_at":"2026-07-05T10:14:16Z"}],"graph_snapshots":[{"event_id":"sha256:47746300fedbf9bb2035eeb5ae55e0dc86c0e9e14ab09af8df1c43bd8ac12a22","target":"graph","created_at":"2026-07-05T10:14:16Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2502.10024/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This paper concerns the study of the incompressible Euler equations with variable density, in the case of space dimension $d=2$. Contrarily to their homogeneous (constant density) counterpart, those equations are not known to be well-posed globally in time. A classical blow-up/continuation criterion for smooth solutions relies on the control of the Lipschitz norm of the velocity field $u$. Here we show that, for establishing blow-up or continuation of solutions, it is enough to determine a control of $\\nabla u$ only along the direction $X=\\nabla^\\perp\\rho$, where $\\rho$ represents the density ","authors_text":"Francesco Fanelli","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-02-14T09:08:23Z","title":"Geometric blow-up criteria for the non-homogeneous incompressible Euler equations in 2-D"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.10024","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":"sha256:423b7d3abd7675a233a07a3f4f26cb5f27824dc53217d294af0126acea714654","target":"record","created_at":"2026-07-05T10:14:16Z","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":"982b4792d9dbb46899ecd4777c3424bb48916a4bb80067185bf2519f815a988e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-02-14T09:08:23Z","title_canon_sha256":"536fefee2156813a1a7d59d4b55578d170ab0b5570519181d4fd1762545cef33"},"schema_version":"1.0","source":{"id":"2502.10024","kind":"arxiv","version":1}},"canonical_sha256":"251ca4dcba1ff219877d6e5c1b1378f9caf72070b36f87f31926f569c9ea06e7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"251ca4dcba1ff219877d6e5c1b1378f9caf72070b36f87f31926f569c9ea06e7","first_computed_at":"2026-07-05T10:14:16.240244Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:14:16.240244Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8K6DHLx2w1xSUx7G2SiZJLadc5ZzPgFddjZcnQltTVlsJjK0z20nINoajKSf+z3uOE2Mcl3MRVKGqHepdGlAAw==","signature_status":"signed_v1","signed_at":"2026-07-05T10:14:16.240738Z","signed_message":"canonical_sha256_bytes"},"source_id":"2502.10024","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:423b7d3abd7675a233a07a3f4f26cb5f27824dc53217d294af0126acea714654","sha256:47746300fedbf9bb2035eeb5ae55e0dc86c0e9e14ab09af8df1c43bd8ac12a22"],"state_sha256":"5924831a9443e73fe235d1c232d65764901ccdf1e1f4a2b15b7f62af10d250ea"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"80GWQKsSv1M5Nfj1l2hKvxUmlmqsODshlDo7zJwifoXGpxEZoqtVTZbySMjIzmYXwaIGqE14CXIOJ6orcMrwBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-15T03:56:06.860285Z","bundle_sha256":"761462eaf6523a5b267f1865e4d2f9d09c8427310c94937ba982f225e91b8fa3"}}