{"paper":{"title":"Global well-posedness and polynomial bounds for the defocusing $L^{2}$-critical nonlinear Schr\\\"odinger equation in $\\R$","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Daniela De Silva, Gigliola Staffilani, Nata\\v{s}a Pavlovi\\'c, Nikolaos Tzirakis","submitted_at":"2007-02-23T16:29:35Z","abstract_excerpt":"We prove global well-posedness for low regularity data for the one dimensional quintic defocusing nonlinear Schr\\\"odinger equation. Precisely we show that a unique and global solution exists for initial data in the Sobolev space $H^{s}(\\mathbb R)$ for any $s>{1/3}$. This improves the result in \\cite{tz}, where global well-posedness was established for any $s>{4/9}$. We use the $I$-method to take advantage of the conservation laws of the equation. The new ingredient in our proof is an interaction Morawetz estimate for the smoothed out solution $Iu$. As a byproduct of our proof we also obtain th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0702707","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"}