{"paper":{"title":"Global solution for the $3D$ quadratic Schr\\\"odinger equation of $Q(u, \\bar{u})$ type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xuecheng Wang","submitted_at":"2016-11-16T02:54:18Z","abstract_excerpt":"We study a class of $3D$ quadratic Schr\\\"odinger equations as follows, $(\\partial_t -i \\Delta) u = Q(u, \\bar{u})$. Different from nonlinearities of the $uu$ type and the $\\bar{u}\\bar{u}$ type, which have been studied by Germain-Masmoudi-Shatah, the interaction of $u$ and $\\bar{u}$ is very strong at the low frequency part, e.g., $1\\times 1 \\rightarrow 0$ type interaction (the size of input frequency is \"$1$\" and the size of output frequency is \"$0$\"). It creates a growth mode for the Fourier transform of the profile of solution around a small neighborhood of zero. This growth mode will again ca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.05124","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"}