{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:L5T7DP2ZP33Q6K53JVWUCYAXDX","short_pith_number":"pith:L5T7DP2Z","canonical_record":{"source":{"id":"1807.04436","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-12T06:28:03Z","cross_cats_sorted":[],"title_canon_sha256":"f7fdb056cfb999cf6359830e4408674cb31ccdc3b4207da555638c360d6804dd","abstract_canon_sha256":"70d532fd72e77ed7732ead826d6892af9c005b5940086daf4debd4ad828a5e60"},"schema_version":"1.0"},"canonical_sha256":"5f67f1bf597ef70f2bbb4d6d4160171de63316d983a1229c546d3ba3762e0a7b","source":{"kind":"arxiv","id":"1807.04436","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.04436","created_at":"2026-05-18T00:10:53Z"},{"alias_kind":"arxiv_version","alias_value":"1807.04436v1","created_at":"2026-05-18T00:10:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.04436","created_at":"2026-05-18T00:10:53Z"},{"alias_kind":"pith_short_12","alias_value":"L5T7DP2ZP33Q","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_16","alias_value":"L5T7DP2ZP33Q6K53","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_8","alias_value":"L5T7DP2Z","created_at":"2026-05-18T12:32:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:L5T7DP2ZP33Q6K53JVWUCYAXDX","target":"record","payload":{"canonical_record":{"source":{"id":"1807.04436","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-12T06:28:03Z","cross_cats_sorted":[],"title_canon_sha256":"f7fdb056cfb999cf6359830e4408674cb31ccdc3b4207da555638c360d6804dd","abstract_canon_sha256":"70d532fd72e77ed7732ead826d6892af9c005b5940086daf4debd4ad828a5e60"},"schema_version":"1.0"},"canonical_sha256":"5f67f1bf597ef70f2bbb4d6d4160171de63316d983a1229c546d3ba3762e0a7b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:53.529676Z","signature_b64":"q8slbWwkqONp+x+8LdBuxulicy9f1KJYc5t70FTZGHJdrGQjYDu6NhzHDDAFwSA3SQ2ZxWohGESloG4dyLqPCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f67f1bf597ef70f2bbb4d6d4160171de63316d983a1229c546d3ba3762e0a7b","last_reissued_at":"2026-05-18T00:10:53.529082Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:53.529082Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1807.04436","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":"2026-05-18T00:10:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1RaZzBZ2Q3cF6J2R5oySEOurfx+B1NvVELEhJ8nby66WJVzGczBJREJ+/pW02LKhJplhNSoQ6p6hA4gRCkPiCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T16:50:58.273447Z"},"content_sha256":"8a772318c0a675234c59472d00377b52d9fe33e76b866a1fd1a53c5df06b4b60","schema_version":"1.0","event_id":"sha256:8a772318c0a675234c59472d00377b52d9fe33e76b866a1fd1a53c5df06b4b60"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:L5T7DP2ZP33Q6K53JVWUCYAXDX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Liouville theorems for the Stokes equations with applications to large time estimates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ken Abe","submitted_at":"2018-07-12T06:28:03Z","abstract_excerpt":"We study Liouville theorems for the non-stationary Stokes equations in exterior domains in $ \\mathbb{R}^{n}$ under decay conditions for spatial variables. As applications, we prove that the Stokes semigroup is a bounded analytic semigroup on $L^{ \\infty}_{ \\sigma}$ of angle $ \\pi/2$ for $n \\geq 3$. We also prove large time estimates for $n=2$ with zero net force."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.04436","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":"2026-05-18T00:10:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ttsPGlwkp094M+XEsYlzsW/wx5OD/Hz8epQIkjNJfbNlQu5O6DbNDL2HfAy7sdZHbaF5WOndxsbYvyV9uXU9Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T16:50:58.273793Z"},"content_sha256":"1f1f9b259f167440535b73dbaae0e9274157d21b2f016a6de3235a63cd8cf65b","schema_version":"1.0","event_id":"sha256:1f1f9b259f167440535b73dbaae0e9274157d21b2f016a6de3235a63cd8cf65b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L5T7DP2ZP33Q6K53JVWUCYAXDX/bundle.json","state_url":"https://pith.science/pith/L5T7DP2ZP33Q6K53JVWUCYAXDX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L5T7DP2ZP33Q6K53JVWUCYAXDX/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":"2026-06-03T16:50:58Z","links":{"resolver":"https://pith.science/pith/L5T7DP2ZP33Q6K53JVWUCYAXDX","bundle":"https://pith.science/pith/L5T7DP2ZP33Q6K53JVWUCYAXDX/bundle.json","state":"https://pith.science/pith/L5T7DP2ZP33Q6K53JVWUCYAXDX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L5T7DP2ZP33Q6K53JVWUCYAXDX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:L5T7DP2ZP33Q6K53JVWUCYAXDX","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":"70d532fd72e77ed7732ead826d6892af9c005b5940086daf4debd4ad828a5e60","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-12T06:28:03Z","title_canon_sha256":"f7fdb056cfb999cf6359830e4408674cb31ccdc3b4207da555638c360d6804dd"},"schema_version":"1.0","source":{"id":"1807.04436","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.04436","created_at":"2026-05-18T00:10:53Z"},{"alias_kind":"arxiv_version","alias_value":"1807.04436v1","created_at":"2026-05-18T00:10:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.04436","created_at":"2026-05-18T00:10:53Z"},{"alias_kind":"pith_short_12","alias_value":"L5T7DP2ZP33Q","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_16","alias_value":"L5T7DP2ZP33Q6K53","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_8","alias_value":"L5T7DP2Z","created_at":"2026-05-18T12:32:33Z"}],"graph_snapshots":[{"event_id":"sha256:1f1f9b259f167440535b73dbaae0e9274157d21b2f016a6de3235a63cd8cf65b","target":"graph","created_at":"2026-05-18T00:10:53Z","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":"We study Liouville theorems for the non-stationary Stokes equations in exterior domains in $ \\mathbb{R}^{n}$ under decay conditions for spatial variables. As applications, we prove that the Stokes semigroup is a bounded analytic semigroup on $L^{ \\infty}_{ \\sigma}$ of angle $ \\pi/2$ for $n \\geq 3$. We also prove large time estimates for $n=2$ with zero net force.","authors_text":"Ken Abe","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-12T06:28:03Z","title":"Liouville theorems for the Stokes equations with applications to large time estimates"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.04436","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":"sha256:8a772318c0a675234c59472d00377b52d9fe33e76b866a1fd1a53c5df06b4b60","target":"record","created_at":"2026-05-18T00:10:53Z","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":"70d532fd72e77ed7732ead826d6892af9c005b5940086daf4debd4ad828a5e60","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-12T06:28:03Z","title_canon_sha256":"f7fdb056cfb999cf6359830e4408674cb31ccdc3b4207da555638c360d6804dd"},"schema_version":"1.0","source":{"id":"1807.04436","kind":"arxiv","version":1}},"canonical_sha256":"5f67f1bf597ef70f2bbb4d6d4160171de63316d983a1229c546d3ba3762e0a7b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5f67f1bf597ef70f2bbb4d6d4160171de63316d983a1229c546d3ba3762e0a7b","first_computed_at":"2026-05-18T00:10:53.529082Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:53.529082Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"q8slbWwkqONp+x+8LdBuxulicy9f1KJYc5t70FTZGHJdrGQjYDu6NhzHDDAFwSA3SQ2ZxWohGESloG4dyLqPCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:53.529676Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.04436","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8a772318c0a675234c59472d00377b52d9fe33e76b866a1fd1a53c5df06b4b60","sha256:1f1f9b259f167440535b73dbaae0e9274157d21b2f016a6de3235a63cd8cf65b"],"state_sha256":"a8831e003aac97ad039fd22a479e65b8b53b6cb6e8e2f10d55312b2ce6f17aa1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MBRrcZNMl2F0qwHKd+r8VkdnTzRwwsEGkjA5x+aDrZ4obGUJK99HMQDLDZt7bejj+lZJGwF9t3MnJIE3b62IBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T16:50:58.275748Z","bundle_sha256":"b5f9336c8bd47318bf6f02c47406a9a44103aecbb7431fef16f15d4ac8ffe2c3"}}