{"paper":{"title":"Continuous Dependence of Cauchy Problem For Nonlinear Schr\\\"{o}dinger Equation in $H^{s}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Daomin Cao, Wei Dai, Weihua Yang","submitted_at":"2010-09-10T13:00:46Z","abstract_excerpt":"We consider the Cauchy problem for the nonlinear Schr\\\"{o}dinger equation $i \\partial_{t}u+ \\Delta u=\\lambda_{0}u+\\lambda_{1}|u|^\\alpha u$ in $\\mathbb{R}^{N}$, where $\\lambda_{0},\\lambda_{1}\\in\\mathbb{C}$, in $H^s$ subcritical and critical case: $0<\\alpha\\leq\\frac{4}{N-2s}$ when $1<s<\\frac{N}{2}$ and $0<\\alpha<+\\infty$ when $s\\geq\\frac{N}{2}$. We show that the solution depends continuously on the initial value in the standard sense in $H^{s}(\\mathbb{R}^{N})$ if $\\alpha$ satisfies certain assumptions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2005","kind":"arxiv","version":4},"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"}