{"title":"超临界表面拟地转方程奇异集维数的估计","authors":"Maria Colombo, Silja Haffter","doi":"10.1007/s40818-021-00093-3","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the SQG equation with dissipation given by a fractional Laplacian of order <span>\\(\\alpha <\\frac{1}{2}\\)</span>. We introduce a notion of suitable weak solution, which exists for every <span>\\(L^2\\)</span> initial datum, and we prove that for such solution the singular set is contained in a compact set in spacetime of Hausdorff dimension at most <span>\\(\\frac{1}{2\\alpha } \\left( \\frac{1+\\alpha }{\\alpha } (1-2\\alpha ) + 2\\right) \\)</span>.</p></div>","PeriodicalId":36382,"journal":{"name":"Annals of Pde","volume":null,"pages":null},"PeriodicalIF":2.4000,"publicationDate":"2021-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40818-021-00093-3","citationCount":"4","resultStr":"{\"title\":\"Estimate on the Dimension of the Singular Set of the Supercritical Surface Quasigeostrophic Equation\",\"authors\":\"Maria Colombo, Silja Haffter\",\"doi\":\"10.1007/s40818-021-00093-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We consider the SQG equation with dissipation given by a fractional Laplacian of order <span>\\\\(\\\\alpha <\\\\frac{1}{2}\\\\)</span>. We introduce a notion of suitable weak solution, which exists for every <span>\\\\(L^2\\\\)</span> initial datum, and we prove that for such solution the singular set is contained in a compact set in spacetime of Hausdorff dimension at most <span>\\\\(\\\\frac{1}{2\\\\alpha } \\\\left( \\\\frac{1+\\\\alpha }{\\\\alpha } (1-2\\\\alpha ) + 2\\\\right) \\\\)</span>.</p></div>\",\"PeriodicalId\":36382,\"journal\":{\"name\":\"Annals of Pde\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.4000,\"publicationDate\":\"2021-03-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1007/s40818-021-00093-3\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annals of Pde\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40818-021-00093-3\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Pde","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s40818-021-00093-3","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Estimate on the Dimension of the Singular Set of the Supercritical Surface Quasigeostrophic Equation
We consider the SQG equation with dissipation given by a fractional Laplacian of order \(\alpha <\frac{1}{2}\). We introduce a notion of suitable weak solution, which exists for every \(L^2\) initial datum, and we prove that for such solution the singular set is contained in a compact set in spacetime of Hausdorff dimension at most \(\frac{1}{2\alpha } \left( \frac{1+\alpha }{\alpha } (1-2\alpha ) + 2\right) \).
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