{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:VRTDYASN7FL54LMOBIXAW3CGML","short_pith_number":"pith:VRTDYASN","canonical_record":{"source":{"id":"1111.2924","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-11-12T13:29:30Z","cross_cats_sorted":[],"title_canon_sha256":"9cea9b5b9f322925544bdf282c11653bcee865eedf1e00f26d5d7678c9503d1c","abstract_canon_sha256":"b3b06155891b20b7590b42f5e323205973d6c2a5483f5afee464822c45131fd8"},"schema_version":"1.0"},"canonical_sha256":"ac663c024df957de2d8e0a2e0b6c4662f55e6a03e899e4259fc14effeb2949ce","source":{"kind":"arxiv","id":"1111.2924","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.2924","created_at":"2026-05-18T03:52:53Z"},{"alias_kind":"arxiv_version","alias_value":"1111.2924v4","created_at":"2026-05-18T03:52:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.2924","created_at":"2026-05-18T03:52:53Z"},{"alias_kind":"pith_short_12","alias_value":"VRTDYASN7FL5","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"VRTDYASN7FL54LMO","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"VRTDYASN","created_at":"2026-05-18T12:26:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:VRTDYASN7FL54LMOBIXAW3CGML","target":"record","payload":{"canonical_record":{"source":{"id":"1111.2924","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-11-12T13:29:30Z","cross_cats_sorted":[],"title_canon_sha256":"9cea9b5b9f322925544bdf282c11653bcee865eedf1e00f26d5d7678c9503d1c","abstract_canon_sha256":"b3b06155891b20b7590b42f5e323205973d6c2a5483f5afee464822c45131fd8"},"schema_version":"1.0"},"canonical_sha256":"ac663c024df957de2d8e0a2e0b6c4662f55e6a03e899e4259fc14effeb2949ce","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:52:53.221756Z","signature_b64":"jpxUOAllq1Wobiv0Uyry6/mcSXQZMhlrXYHwuw9FphP11YcCckij939OMVy70CWGRNIlxBPhyueGJhqJezeBBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac663c024df957de2d8e0a2e0b6c4662f55e6a03e899e4259fc14effeb2949ce","last_reissued_at":"2026-05-18T03:52:53.220821Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:52:53.220821Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.2924","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:52:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gAaRgXGNQ9pEunuXqW6yQFOBDJQHkDs4z8dtapv9/Y4XA+2w+zIAqrizQd85lRP30MslVFv9aIPiHDSpgNJ4Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T02:04:28.105973Z"},"content_sha256":"5d34990203ed1c1602423ad66447966ed1c70200d0755bd85678873e9dd257bc","schema_version":"1.0","event_id":"sha256:5d34990203ed1c1602423ad66447966ed1c70200d0755bd85678873e9dd257bc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:VRTDYASN7FL54LMOBIXAW3CGML","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Trajectory attractor for a non-autonomous Magnetohydrodynamic equations of Non-Newtonian Fluids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Paul Andre Razafimandimby","submitted_at":"2011-11-12T13:29:30Z","abstract_excerpt":"In this article we initiate the mathematical study of the dynamics of a system of nonlinear Partial Differential Equations modelling the motion of incompressible, isothermal and conducting modified bipolar fluids in presence of magnetic field. We mainly prove the existence of weak solutions to the model. We also prove the existence of a trajectory attractor to the translation semigroup acting on the trajectories of the set of weak solutions and that of external forces. Some results concerning the structure of this trajectory attractor are also given. The results from this paper may be useful i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.2924","kind":"arxiv","version":4},"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":"2026-05-18T03:52:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZN3KIu6swMX7LaEbEcuNd/nl6WbCTZ3zUUN0IUPwq7+Eghp0eMkQpfnG4SJYWr9M+CDXmd6k5zu1zpiVwjdoCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T02:04:28.106357Z"},"content_sha256":"c45b3d975008b5d19d406ad75dcf9efa799cacc590902272f5a73db9482a0f56","schema_version":"1.0","event_id":"sha256:c45b3d975008b5d19d406ad75dcf9efa799cacc590902272f5a73db9482a0f56"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VRTDYASN7FL54LMOBIXAW3CGML/bundle.json","state_url":"https://pith.science/pith/VRTDYASN7FL54LMOBIXAW3CGML/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VRTDYASN7FL54LMOBIXAW3CGML/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":"2026-06-27T02:04:28Z","links":{"resolver":"https://pith.science/pith/VRTDYASN7FL54LMOBIXAW3CGML","bundle":"https://pith.science/pith/VRTDYASN7FL54LMOBIXAW3CGML/bundle.json","state":"https://pith.science/pith/VRTDYASN7FL54LMOBIXAW3CGML/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VRTDYASN7FL54LMOBIXAW3CGML/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:VRTDYASN7FL54LMOBIXAW3CGML","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":"b3b06155891b20b7590b42f5e323205973d6c2a5483f5afee464822c45131fd8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-11-12T13:29:30Z","title_canon_sha256":"9cea9b5b9f322925544bdf282c11653bcee865eedf1e00f26d5d7678c9503d1c"},"schema_version":"1.0","source":{"id":"1111.2924","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.2924","created_at":"2026-05-18T03:52:53Z"},{"alias_kind":"arxiv_version","alias_value":"1111.2924v4","created_at":"2026-05-18T03:52:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.2924","created_at":"2026-05-18T03:52:53Z"},{"alias_kind":"pith_short_12","alias_value":"VRTDYASN7FL5","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"VRTDYASN7FL54LMO","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"VRTDYASN","created_at":"2026-05-18T12:26:44Z"}],"graph_snapshots":[{"event_id":"sha256:c45b3d975008b5d19d406ad75dcf9efa799cacc590902272f5a73db9482a0f56","target":"graph","created_at":"2026-05-18T03:52:53Z","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":"In this article we initiate the mathematical study of the dynamics of a system of nonlinear Partial Differential Equations modelling the motion of incompressible, isothermal and conducting modified bipolar fluids in presence of magnetic field. We mainly prove the existence of weak solutions to the model. We also prove the existence of a trajectory attractor to the translation semigroup acting on the trajectories of the set of weak solutions and that of external forces. Some results concerning the structure of this trajectory attractor are also given. The results from this paper may be useful i","authors_text":"Paul Andre Razafimandimby","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-11-12T13:29:30Z","title":"Trajectory attractor for a non-autonomous Magnetohydrodynamic equations of Non-Newtonian Fluids"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.2924","kind":"arxiv","version":4},"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":"sha256:5d34990203ed1c1602423ad66447966ed1c70200d0755bd85678873e9dd257bc","target":"record","created_at":"2026-05-18T03:52:53Z","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":"b3b06155891b20b7590b42f5e323205973d6c2a5483f5afee464822c45131fd8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-11-12T13:29:30Z","title_canon_sha256":"9cea9b5b9f322925544bdf282c11653bcee865eedf1e00f26d5d7678c9503d1c"},"schema_version":"1.0","source":{"id":"1111.2924","kind":"arxiv","version":4}},"canonical_sha256":"ac663c024df957de2d8e0a2e0b6c4662f55e6a03e899e4259fc14effeb2949ce","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ac663c024df957de2d8e0a2e0b6c4662f55e6a03e899e4259fc14effeb2949ce","first_computed_at":"2026-05-18T03:52:53.220821Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:52:53.220821Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jpxUOAllq1Wobiv0Uyry6/mcSXQZMhlrXYHwuw9FphP11YcCckij939OMVy70CWGRNIlxBPhyueGJhqJezeBBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:52:53.221756Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.2924","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5d34990203ed1c1602423ad66447966ed1c70200d0755bd85678873e9dd257bc","sha256:c45b3d975008b5d19d406ad75dcf9efa799cacc590902272f5a73db9482a0f56"],"state_sha256":"09bb4379a52612148a27b198d6672fd6902384f503c5c3a4a2c7d91a7a2fa7c2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XqIWdD5Mu8lRI+4LfcjnAvcX/+1qsMdCJcpZez/FZAebQxPZDmakIaGGxkGYH5qqYAIjw5A4KMHgECN8aFF+Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T02:04:28.108423Z","bundle_sha256":"b83de9e56fee5e3c48b39631ecf9a94fa8b1c7eb7ab53c7d95a200b75b6437bf"}}