{"title":"Homogenization of fundamental solutions for parabolic operators involving non-self-similar scales","authors":"Qing Meng, Weisheng Niu","doi":"10.1007/s10231-024-01446-y","DOIUrl":null,"url":null,"abstract":"<div><p>We establish the asymptotic expansion of the fundamental solutions with precise error estimates for second-order parabolic operators </p><div><div><span>$$\\begin{aligned} \\partial _t -\\text {div}(A(x/\\varepsilon , t/\\varepsilon ^\\ell )\\nabla ), \\quad \\, 0<\\varepsilon<1,\\, 0<\\ell <\\infty ,\\end{aligned}$$</span></div></div><p>in the case <span>\\(\\ell \\ne 2,\\)</span> where the spatial and temporal variables oscillate on non-self-similar scales and do not homogenize simultaneously. To achieve the goal, we explore the direct quantitative two-scale expansions for the aforementioned operators, which should be of some independent interests in quantitative homogenization of parabolic operators involving multiple scales. In the self-similar case <span>\\(\\ell =2\\)</span>, similar results have been obtained in Geng and Shen (Anal PDE 13(1): 147–170, 2020).</p></div>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annali di Matematica Pura ed Applicata","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10231-024-01446-y","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We establish the asymptotic expansion of the fundamental solutions with precise error estimates for second-order parabolic operators
in the case \(\ell \ne 2,\) where the spatial and temporal variables oscillate on non-self-similar scales and do not homogenize simultaneously. To achieve the goal, we explore the direct quantitative two-scale expansions for the aforementioned operators, which should be of some independent interests in quantitative homogenization of parabolic operators involving multiple scales. In the self-similar case \(\ell =2\), similar results have been obtained in Geng and Shen (Anal PDE 13(1): 147–170, 2020).
期刊介绍:
This journal, the oldest scientific periodical in Italy, was originally edited by Barnaba Tortolini and Francesco Brioschi and has appeared since 1850. Nowadays it is managed by a nonprofit organization, the Fondazione Annali di Matematica Pura ed Applicata, c.o. Dipartimento di Matematica "U. Dini", viale Morgagni 67A, 50134 Firenze, Italy, e-mail annali@math.unifi.it).
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