{"paper":{"title":"Global well-posedness for the defocusing Hartree equation with radial data in $\\mathbb R^4$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Changxing Miao, Guixiang Xu, Jianwei Yang","submitted_at":"2018-07-07T09:41:06Z","abstract_excerpt":"By $I$-method, the interaction Morawetz estimate, long time Strichartz estimate and local smoothing effect of Schr\\\"odinger operator, we show global well-posedness and scattering for the defocusing Hartree equation  $$\\left\\{ \\begin{array}{ll} iu_t + \\Delta u &=F(u), \\quad (t,x) \\in \\mathbb{R} \\times \\mathbb{R}^4 u(0) \\\\ &=u_0(x)\\in H^s(\\mathbb{R}^4), \\end{array} \\right. $$ where $F(u)= (V* |u|^2) u$, and $V(x)=|x|^{-\\gamma}$, $3< \\gamma<4$, with radial data in $H^{s}(\\mathbb{R}^4)$ for $s>s_c:=\\gamma/2-1$. It is a sharp global result except of the critical case $s=s_c$, which is a very diffic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.05841","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1807.05841/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}