{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:PVOTSI63XZVCE5BCYPDEHQPRGA","short_pith_number":"pith:PVOTSI63","schema_version":"1.0","canonical_sha256":"7d5d3923dbbe6a227422c3c643c1f13034d5e10cf112219fd17d1bf82b2f3b8c","source":{"kind":"arxiv","id":"1205.5188","version":1},"attestation_state":"computed","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 $01$. 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