{"title":"Global well-posedness and scattering of 3D defocusing, cubic Schrödinger equation","authors":"Jia Shen, Yifei Wu","doi":"10.4310/mrl.2023.v30.n6.a10","DOIUrl":null,"url":null,"abstract":"In this paper, we study the global well-posedness and scattering of 3D defocusing, cubic Schrödinger equation. Recently, Dodson $\\href{https://dx.doi.org/10.4171/RMI/1295}{\\textrm{[16]}}$ studied the global well-posedness in a critical Sobolev space $\\dot{W}^{11/7,7/6}$. In this paper, we aim to show that if the initial data belongs to $\\dot{H}^{\\frac{1}{2}}$ to guarantee the local existence, then some extra weak space which is supercritical, is sufficient to prove the global well-posedness. More precisely, we prove that if the initial data belongs to $\\dot{H}^{1/2} \\cap \\dot{W}^{s,1}$ for $12/13 \\lt s \\leqslant 1$, then the corresponding solution exists globally and scatters.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"23 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Research Letters","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/mrl.2023.v30.n6.a10","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we study the global well-posedness and scattering of 3D defocusing, cubic Schrödinger equation. Recently, Dodson $\href{https://dx.doi.org/10.4171/RMI/1295}{\textrm{[16]}}$ studied the global well-posedness in a critical Sobolev space $\dot{W}^{11/7,7/6}$. In this paper, we aim to show that if the initial data belongs to $\dot{H}^{\frac{1}{2}}$ to guarantee the local existence, then some extra weak space which is supercritical, is sufficient to prove the global well-posedness. More precisely, we prove that if the initial data belongs to $\dot{H}^{1/2} \cap \dot{W}^{s,1}$ for $12/13 \lt s \leqslant 1$, then the corresponding solution exists globally and scatters.
期刊介绍:
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