{"paper":{"title":"Uniqueness and long time asymptotic for the Keller-Segel equation: The parabolic-elliptic case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fernandez Giani Egana, St\\'ephane Mischler (CEREMADE)","submitted_at":"2013-10-29T12:01:37Z","abstract_excerpt":"The present paper deals with the parabolic-elliptic Keller-Segel equation in the plane in the general framework of weak (or ''free energy\") solutions associated to initial datum with finite mass $M$, finite second moment and finite entropy. The aim of the paper is threefold: (1) We prove the uniqueness of the ''free energy\" solution on the maximal interval of existence $[0,T^*)$ with $T^*=\\infty$ in the case when $M\\le8\\pi$ and $T^*< \\infty$ in the case when $M> 8\\pi$. The proof uses a DiPerna-Lions renormalizing argument which makes possible to get the ''optimal regularity\" as well as an esti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.7771","kind":"arxiv","version":2},"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"}