{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:7OLUWPOVW5RVYGRP3RVX73VSF2","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":"528a20ac6ed64e4b2461164ae20b477499f86df339b59c6c333fda6a06ecffc7","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-05-04T07:22:36Z","title_canon_sha256":"995bee53abeff2e95a71a045bd8aa6e5f5594e89db1bff5871557fba88d429da"},"schema_version":"1.0","source":{"id":"1805.01630","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.01630","created_at":"2026-05-18T00:16:47Z"},{"alias_kind":"arxiv_version","alias_value":"1805.01630v1","created_at":"2026-05-18T00:16:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.01630","created_at":"2026-05-18T00:16:47Z"},{"alias_kind":"pith_short_12","alias_value":"7OLUWPOVW5RV","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"7OLUWPOVW5RVYGRP","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"7OLUWPOV","created_at":"2026-05-18T12:32:11Z"}],"graph_snapshots":[{"event_id":"sha256:70443583fa0cdf43987e7300ffd11e7c4a67115727096f9830361132843b5793","target":"graph","created_at":"2026-05-18T00:16:47Z","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 show that, if $b\\in L^1(0,T;L^1_{\\mathrm{loc}}(\\mathbb{R}))$ has spatial derivative in the John-Nirenberg space $\\mathrm{BMO}(\\mathbb{R})$, then it generalizes a unique flow $\\phi(t,\\cdot)$ which has an $A_\\infty(\\mathbb R)$ density for each time $t\\in [0,T]$. Our condition on the map $b$ is optimal and we also get a sharp quantitative estimate for the density. As a natural application we establish a well-posedness for the Cauchy problem of the transport equation in $\\mathrm{BMO}(\\mathbb R)$.","authors_text":"Jie Xiao, Kangwei Li, Renjin Jiang","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-05-04T07:22:36Z","title":"Flow with $A_\\infty(\\mathbb R)$ density and transport equation in $\\mathrm{BMO}(\\mathbb R)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.01630","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":"sha256:067061b9b0bf7cd10b053fac72f187c40d046d037b5401106916a917ac848b53","target":"record","created_at":"2026-05-18T00:16:47Z","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":"528a20ac6ed64e4b2461164ae20b477499f86df339b59c6c333fda6a06ecffc7","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-05-04T07:22:36Z","title_canon_sha256":"995bee53abeff2e95a71a045bd8aa6e5f5594e89db1bff5871557fba88d429da"},"schema_version":"1.0","source":{"id":"1805.01630","kind":"arxiv","version":1}},"canonical_sha256":"fb974b3dd5b7635c1a2fdc6b7feeb22e9315e8cf85d9a2ab1eb0dcd6ebdd6e87","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fb974b3dd5b7635c1a2fdc6b7feeb22e9315e8cf85d9a2ab1eb0dcd6ebdd6e87","first_computed_at":"2026-05-18T00:16:47.244345Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:16:47.244345Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gLL1AYgT4blKFcda/y9cjtPwZagn3t3NgdjmDMQRrh8u64uVk8nEJj0Qt3dDmBNr9zj8239vUbQiQwXqAT7qCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:16:47.245100Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.01630","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:067061b9b0bf7cd10b053fac72f187c40d046d037b5401106916a917ac848b53","sha256:70443583fa0cdf43987e7300ffd11e7c4a67115727096f9830361132843b5793"],"state_sha256":"64311b938985b3e8c226fe36d35e3686f1318cab9cfae9e90879263c4950ceb9"}