{"paper":{"title":"Remark on the periodic mass critical nonlinear Schr\\\"odinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nobu Kishimoto","submitted_at":"2012-03-28T20:43:04Z","abstract_excerpt":"We consider the mass critical NLS on $\\mathbb{T}$ and $\\mathbb{T}^2$. In the $\\mathbb{R}^d$ case the Strichartz estimates enable us to show well-posedness of the IVP in $L^2$ (at least for small data) via the Picard iteration method. However, counterexamples to the $L^6$ Strichartz on $\\mathbb{T}$ and the $L^4$ Strichartz on $\\mathbb{T}^2$ were given by Bourgain (1993) and Takaoka-Tzvetkov (2001), respectively, which means that the Strichartz spaces are not suitable for iteration in these problems. In this note, we show a slightly stronger result, namely, that the IVP on $\\mathbb{T}$ and $\\mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.6375","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"}