{"paper":{"title":"Flow with $A_\\infty(\\mathbb R)$ density and transport equation in $\\mathrm{BMO}(\\mathbb R)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Jie Xiao, Kangwei Li, Renjin Jiang","submitted_at":"2018-05-04T07:22:36Z","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 $\\mathrm{BMO}(\\mathbb R)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.01630","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"}