{"title":"Local well-posedness for the Zakharov system in dimension d ≤ 3","authors":"A. Sanwal","doi":"10.3934/dcds.2021147","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>The Zakharov system in dimension <inline-formula><tex-math id=\"M1\">\\begin{document}$ d\\leqslant 3 $\\end{document}</tex-math></inline-formula> is shown to be locally well-posed in Sobolev spaces <inline-formula><tex-math id=\"M2\">\\begin{document}$ H^s \\times H^l $\\end{document}</tex-math></inline-formula>, extending the previously known result. We construct new solution spaces by modifying the <inline-formula><tex-math id=\"M3\">\\begin{document}$ X^{s,b} $\\end{document}</tex-math></inline-formula> spaces, specifically by introducing temporal weights. We use the contraction mapping principle to prove local well-posedness in the same. The result obtained is sharp up to endpoints.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"28 10 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2021-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete & Continuous Dynamical Systems - S","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/dcds.2021147","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 5
Abstract
The Zakharov system in dimension \begin{document}$ d\leqslant 3 $\end{document} is shown to be locally well-posed in Sobolev spaces \begin{document}$ H^s \times H^l $\end{document}, extending the previously known result. We construct new solution spaces by modifying the \begin{document}$ X^{s,b} $\end{document} spaces, specifically by introducing temporal weights. We use the contraction mapping principle to prove local well-posedness in the same. The result obtained is sharp up to endpoints.
The Zakharov system in dimension \begin{document}$ d\leqslant 3 $\end{document} is shown to be locally well-posed in Sobolev spaces \begin{document}$ H^s \times H^l $\end{document}, extending the previously known result. We construct new solution spaces by modifying the \begin{document}$ X^{s,b} $\end{document} spaces, specifically by introducing temporal weights. We use the contraction mapping principle to prove local well-posedness in the same. The result obtained is sharp up to endpoints.
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