{"paper":{"title":"Uniqueness and long time asymptotics for the parabolic-parabolic Keller-Segel equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kleber Carrapatoso (CMLA), St\\'ephane Mischler (CEREMADE)","submitted_at":"2014-06-23T17:48:11Z","abstract_excerpt":"The present paper deals with the parabolic-parabolic Keller-Segel equation in the plane inthe general framework of weak (or \"free energy\") solutions associated to an initial datum with finite mass $M\\textless{} 8\\pi$,  finite second log-moment and finite entropy. The aim of the paper is twofold:(1) We prove the uniqueness of the \"free energy\" solution. The proof uses a DiPerna-Lions renormalizing argument which makes possible to get the \"optimal regularity\" as well as an estimate of the difference of two possible solutions in the critical $L^{4/3}$ Lebesgue norm similarly as for the $2d$  vort"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.6006","kind":"arxiv","version":3},"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"}