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In particular, our results can be applied to elliptic operator $L=\\nu\\Delta+u(x,t)\\cdot\\nabla$, where $u(\\cdot,t)$ is a time-dependent vector field in $\\mathbb{R}^{n}$, which is divergence-free in distribution sense, i.e. $\\nabla\\cdot u=0$. Suppose $u\\in L_{t}^{\\infty}(\\textrm{BMO}_{x}^{-1})$. We show the existence of the fundamental solution $\\varGamma(x,t;\\xi,\\tau)$ of the parabolic operator $L-\\partial_{t}$, and show that $\\varGamma$ satisfies the Aronson estimate with a constant depending only on t"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1612.07727","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-22T17:54:30Z","cross_cats_sorted":[],"title_canon_sha256":"3017c8bfbdbd4c366dcc1371f66ad18382ed79a6bead1b7e34befa6785f4c9ef","abstract_canon_sha256":"2fd6909a972d6a0fe5338d973f78ad06ece298b344ac538a96ab9430af6f1936"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:05.619758Z","signature_b64":"fr27BdQwC3I01Au4eRfsL3JhPQz3+hkklfaUAYC51S+sRDGWXul1AT22Q0unicxYwJ970S7ebncYLpzMW3ZzAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d1447a572fc4866b8ea5f3f7282184cb442742c450f44cd66217f7bc77fcdc39","last_reissued_at":"2026-05-17T23:58:05.619295Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:05.619295Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Parabolic equations with singular divergence-free drift vector fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guangyu Xi, Zhongmin Qian","submitted_at":"2016-12-22T17:54:30Z","abstract_excerpt":"In this paper, we study an elliptic operator in divergence-form but not necessary symmetric. In particular, our results can be applied to elliptic operator $L=\\nu\\Delta+u(x,t)\\cdot\\nabla$, where $u(\\cdot,t)$ is a time-dependent vector field in $\\mathbb{R}^{n}$, which is divergence-free in distribution sense, i.e. $\\nabla\\cdot u=0$. Suppose $u\\in L_{t}^{\\infty}(\\textrm{BMO}_{x}^{-1})$. 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