{"title":"Partial Regularity of Suitable Weak Solutions of the Model Arising in Amorphous Molecular Beam Epitaxy","authors":"Yan Qing Wang, Yi Ke Huang, Gang Wu, Dao Guo Zhou","doi":"10.1007/s10114-023-2458-2","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set <span>\\({\\cal S}\\)</span> of suitable weak solutions and the parameter <i>α</i> in the nonlinear term in the following parabolic equation </p><div><div><span>$${h_t} + {h_{xxxx}} + {\\partial _{xx}}|{h_x}{|^\\alpha } = f.$$</span></div></div><p> It is shown that when <span>\\(5/3 \\le \\alpha < 7/3\\)</span>, the <span>\\({{3\\alpha - 5} \\over {\\alpha - 1}}\\)</span> dimensional parabolic Hausdorff measure of <span>\\({\\cal S}\\)</span> is zero, which generalizes the recent corresponding work of Ozánski and Robinson in [<i>SIAM J. Math. Anal.</i>, <b>51</b>, 228–255 (2019)] for <i>α</i> = 2 and <i>f</i> = 0. The same result is valid for a 3D modified Navier–Stokes system.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10114-023-2458-2.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Sinica-English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10114-023-2458-2","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set \({\cal S}\) of suitable weak solutions and the parameter α in the nonlinear term in the following parabolic equation
It is shown that when \(5/3 \le \alpha < 7/3\), the \({{3\alpha - 5} \over {\alpha - 1}}\) dimensional parabolic Hausdorff measure of \({\cal S}\) is zero, which generalizes the recent corresponding work of Ozánski and Robinson in [SIAM J. Math. Anal., 51, 228–255 (2019)] for α = 2 and f = 0. The same result is valid for a 3D modified Navier–Stokes system.
期刊介绍:
Acta Mathematica Sinica, established by the Chinese Mathematical Society in 1936, is the first and the best mathematical journal in China. In 1985, Acta Mathematica Sinica is divided into English Series and Chinese Series. The English Series is a monthly journal, publishing significant research papers from all branches of pure and applied mathematics. It provides authoritative reviews of current developments in mathematical research. Contributions are invited from researchers from all over the world.