{"paper":{"title":"Growth of Sobolev norms in the cubic defocusing nonlinear Schr\\\"odinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.AP","authors_text":"Marcel Guardia, Vadim Kaloshin","submitted_at":"2012-05-23T14:41:00Z","abstract_excerpt":"We consider the cubic defocusing nonlinear Schr\\\"odinger equation in the two dimensional torus. Fix $s>1$. Colliander, Keel, Staffilani, Tao and Takaoka proved in \\cite{CollianderKSTT10} the existence of solutions with $s$-Sobolev norm growing in time.\n  We establish the existence of solutions with polynomial time estimates. More exactly, there is $c>0$ such that for any $\\mathcal{K}\\gg 1$ we find a solution $u$ and a time $T$ such that $\\| u(T)\\|_{H^s}\\geq\\mathcal{K} \\| u(0)\\|_{H^s}$. Moreover, time $T$ satisfies polynomial bound $0<T<\\mathcal{K}^c$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.5188","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"}