{"paper":{"title":"Global well-posedness for Schr\\\"odinger equation with derivative in $H^{{1/2}}(\\R)$","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Changxing Miao, Guixiang Xu, Yifei Wu","submitted_at":"2009-12-23T13:51:48Z","abstract_excerpt":"In this paper, we consider the Cauchy problem of the cubic nonlinear Schr\\\"{o}dinger equation with derivative in $H^s(\\R)$. This equation was known to be the local well-posedness for $s\\geq \\frac12$ (Takaoka,1999), ill-posedness for $s<\\frac12$ (Biagioni and Linares, 2001, etc.) and global well-posedness for $s>\\frac12$ (I-team, 2002). In this paper, we show that it is global well-posedness in $H^{1/2(\\R)$. The main approach is the third generation I-method combined with some additional resonant decomposition technique. The resonant decomposition is applied to control the singularity coming fr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.4642","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"}