{"paper":{"title":"Global Well-Posedness for a periodic nonlinear Schr\\\"odinger equation in 1D and 2D","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":"2006-02-24T20:49:32Z","abstract_excerpt":"The initial value problem for the $L^{2}$ critical semilinear Schr\\\"odinger equation with periodic boundary data is considered. We show that the problem is globally well posed in $H^{s}({\\Bbb T^{d}})$, for $s>4/9$ and $s>2/3$ in 1D and 2D respectively, confirming in 2D a statement of Bourgain in \\cite{bo2}. We use the ``$I$-method''. This method allows one to introduce a modification of the energy functional that is well defined for initial data below the $H^{1}({\\Bbb T^{d}})$ threshold. The main ingredient in the proof is a \"refinement\" of the Strichartz's estimates that hold true for solutio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0602560","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":""},"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"}