{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:ZZUGPNQDFEWDV7WLIDIZEOCXVM","short_pith_number":"pith:ZZUGPNQD","canonical_record":{"source":{"id":"1602.03193","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-09T21:02:33Z","cross_cats_sorted":[],"title_canon_sha256":"83c73b9ee7be869601a63a11a8e980c9f1b80d5a77114afdf02759a48b7494b3","abstract_canon_sha256":"5dd86db18824eea66006b83a03ff7ecfb2893a9acc9c26d8188c71d05b5de049"},"schema_version":"1.0"},"canonical_sha256":"ce6867b603292c3afecb40d1923857ab19aba385a805caeca98a6125bbf0d245","source":{"kind":"arxiv","id":"1602.03193","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.03193","created_at":"2026-05-18T01:21:00Z"},{"alias_kind":"arxiv_version","alias_value":"1602.03193v1","created_at":"2026-05-18T01:21:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.03193","created_at":"2026-05-18T01:21:00Z"},{"alias_kind":"pith_short_12","alias_value":"ZZUGPNQDFEWD","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"ZZUGPNQDFEWDV7WL","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"ZZUGPNQD","created_at":"2026-05-18T12:30:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:ZZUGPNQDFEWDV7WLIDIZEOCXVM","target":"record","payload":{"canonical_record":{"source":{"id":"1602.03193","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-09T21:02:33Z","cross_cats_sorted":[],"title_canon_sha256":"83c73b9ee7be869601a63a11a8e980c9f1b80d5a77114afdf02759a48b7494b3","abstract_canon_sha256":"5dd86db18824eea66006b83a03ff7ecfb2893a9acc9c26d8188c71d05b5de049"},"schema_version":"1.0"},"canonical_sha256":"ce6867b603292c3afecb40d1923857ab19aba385a805caeca98a6125bbf0d245","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:00.568244Z","signature_b64":"AwYWs4td5wBcUMil80jryiJnD0mL6oMX9ShsbwCxw1qbeve/0M1Eb158sQcU1ntiknvAgQx62DWNr0569GZfCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ce6867b603292c3afecb40d1923857ab19aba385a805caeca98a6125bbf0d245","last_reissued_at":"2026-05-18T01:21:00.567784Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:00.567784Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.03193","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":"2026-05-18T01:21:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"t3ZcxhamUuEzDQ4nDtBUDImVbUUQDKfLYg2iL/82fUjXpmn5zj1lAz7whFKGN4wHRAl1bW2VYDxNIW9uD6nNAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T22:15:15.362475Z"},"content_sha256":"6707cedf5034fefa6e1d7371d1bbbf5832d9b8f6080940cae03881001a1c37b0","schema_version":"1.0","event_id":"sha256:6707cedf5034fefa6e1d7371d1bbbf5832d9b8f6080940cae03881001a1c37b0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:ZZUGPNQDFEWDV7WLIDIZEOCXVM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Transport equation with integral terms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Agnieszka \\'Swierczewska-Gwiazda, Camillo De Lellis, Piotr Gwiazda","submitted_at":"2016-02-09T21:02:33Z","abstract_excerpt":"We prove some theorems on the existence, uniqueness, stability and compactness properties of solutions to inhomogeneous transport equations with Sobolev coefficients, where the inhomogeneous term depends upon the solution through an integral operator. Contrary to the usual DiPerna-Lions approach, the essential step is to formulate the problem in the Lagrangian setting. Some motivations to study the above problem arise from the description of polymeric flows, where such kind of equations are coupled with other Navier-Stokes type equations. Using the results for the transport equation we will pr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.03193","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":"2026-05-18T01:21:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QD5HOk53KD+0WhcpLZhMsrXgInCJIshJheLYRMdS9qcobz6f7rS/CMVDuujVH9qWoYWxiZZGpCjuS28HaCZaCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T22:15:15.363087Z"},"content_sha256":"cae3142b20d1fe2a3d1658e500cc89aeb4abcb919490818997de53aa53db9754","schema_version":"1.0","event_id":"sha256:cae3142b20d1fe2a3d1658e500cc89aeb4abcb919490818997de53aa53db9754"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZZUGPNQDFEWDV7WLIDIZEOCXVM/bundle.json","state_url":"https://pith.science/pith/ZZUGPNQDFEWDV7WLIDIZEOCXVM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZZUGPNQDFEWDV7WLIDIZEOCXVM/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":"2026-05-25T22:15:15Z","links":{"resolver":"https://pith.science/pith/ZZUGPNQDFEWDV7WLIDIZEOCXVM","bundle":"https://pith.science/pith/ZZUGPNQDFEWDV7WLIDIZEOCXVM/bundle.json","state":"https://pith.science/pith/ZZUGPNQDFEWDV7WLIDIZEOCXVM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZZUGPNQDFEWDV7WLIDIZEOCXVM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:ZZUGPNQDFEWDV7WLIDIZEOCXVM","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":"5dd86db18824eea66006b83a03ff7ecfb2893a9acc9c26d8188c71d05b5de049","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-09T21:02:33Z","title_canon_sha256":"83c73b9ee7be869601a63a11a8e980c9f1b80d5a77114afdf02759a48b7494b3"},"schema_version":"1.0","source":{"id":"1602.03193","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.03193","created_at":"2026-05-18T01:21:00Z"},{"alias_kind":"arxiv_version","alias_value":"1602.03193v1","created_at":"2026-05-18T01:21:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.03193","created_at":"2026-05-18T01:21:00Z"},{"alias_kind":"pith_short_12","alias_value":"ZZUGPNQDFEWD","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"ZZUGPNQDFEWDV7WL","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"ZZUGPNQD","created_at":"2026-05-18T12:30:55Z"}],"graph_snapshots":[{"event_id":"sha256:cae3142b20d1fe2a3d1658e500cc89aeb4abcb919490818997de53aa53db9754","target":"graph","created_at":"2026-05-18T01:21:00Z","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 prove some theorems on the existence, uniqueness, stability and compactness properties of solutions to inhomogeneous transport equations with Sobolev coefficients, where the inhomogeneous term depends upon the solution through an integral operator. Contrary to the usual DiPerna-Lions approach, the essential step is to formulate the problem in the Lagrangian setting. Some motivations to study the above problem arise from the description of polymeric flows, where such kind of equations are coupled with other Navier-Stokes type equations. Using the results for the transport equation we will pr","authors_text":"Agnieszka \\'Swierczewska-Gwiazda, Camillo De Lellis, Piotr Gwiazda","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-09T21:02:33Z","title":"Transport equation with integral terms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.03193","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":"sha256:6707cedf5034fefa6e1d7371d1bbbf5832d9b8f6080940cae03881001a1c37b0","target":"record","created_at":"2026-05-18T01:21:00Z","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":"5dd86db18824eea66006b83a03ff7ecfb2893a9acc9c26d8188c71d05b5de049","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-09T21:02:33Z","title_canon_sha256":"83c73b9ee7be869601a63a11a8e980c9f1b80d5a77114afdf02759a48b7494b3"},"schema_version":"1.0","source":{"id":"1602.03193","kind":"arxiv","version":1}},"canonical_sha256":"ce6867b603292c3afecb40d1923857ab19aba385a805caeca98a6125bbf0d245","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ce6867b603292c3afecb40d1923857ab19aba385a805caeca98a6125bbf0d245","first_computed_at":"2026-05-18T01:21:00.567784Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:21:00.567784Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AwYWs4td5wBcUMil80jryiJnD0mL6oMX9ShsbwCxw1qbeve/0M1Eb158sQcU1ntiknvAgQx62DWNr0569GZfCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:21:00.568244Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.03193","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6707cedf5034fefa6e1d7371d1bbbf5832d9b8f6080940cae03881001a1c37b0","sha256:cae3142b20d1fe2a3d1658e500cc89aeb4abcb919490818997de53aa53db9754"],"state_sha256":"9b29303c0ae825db34076e87de12a3f653fc189a00ef4e705a64ddd61ed38d49"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"isNbfbtlkkYfaTZ7QwJ2JIU16KcyrZB6BHWxGvjTPVS3khRqyy4pW4Hw1duX8p6TI61e8e0MF5USM2NOg9VnAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T22:15:15.365517Z","bundle_sha256":"478cd9b3d5c0c4a14103002c25d3d27e5e65b18449f2fe9b1ca50c0ca2df1f54"}}