{"title":"Variations on the theme of the Trotter-Kato theorem for homogenization of periodic hyperbolic systems","authors":"Yu.M. Meshkova","doi":"10.1134/s106192082304012x","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> In <span>\\(L_2(\\mathbb{R}^d;\\mathbb{C}^n)\\)</span>, we consider a matrix elliptic second order differential operator <span>\\(B_\\varepsilon >0\\)</span>. Coefficients of the operator <span>\\(B_\\varepsilon\\)</span> are periodic with respect to some lattice in <span>\\(\\mathbb{R}^d\\)</span> and depend on <span>\\(\\mathbf{x}/\\varepsilon\\)</span>. We study the quantitative homogenization for the solutions of the hyperbolic system <span>\\(\\partial _t^2\\mathbf{u}_\\varepsilon =-B_\\varepsilon\\mathbf{u}_\\varepsilon\\)</span>. In operator terms, we are interested in approximations of the operators <span>\\(\\cos (tB_\\varepsilon ^{1/2})\\)</span> and <span>\\(B_\\varepsilon ^{-1/2}\\sin (tB_\\varepsilon ^{1/2})\\)</span> in suitable operator norms. Approximations for the resolvent <span>\\(B_\\varepsilon ^{-1}\\)</span> have been already obtained by T.A. Suslina. So, we rewrite hyperbolic equation as a system for the vector with components <span>\\(\\mathbf{u}_\\varepsilon \\)</span> and <span>\\(\\partial _t\\mathbf{u}_\\varepsilon\\)</span>, and consider the corresponding unitary group. For this group, we adapt the proof of the Trotter-Kato theorem by introduction of some correction term and derive hyperbolic results from elliptic ones. </p><p> <b> DOI</b> 10.1134/S106192082304012X </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.7000,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Journal of Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1134/s106192082304012x","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
In \(L_2(\mathbb{R}^d;\mathbb{C}^n)\), we consider a matrix elliptic second order differential operator \(B_\varepsilon >0\). Coefficients of the operator \(B_\varepsilon\) are periodic with respect to some lattice in \(\mathbb{R}^d\) and depend on \(\mathbf{x}/\varepsilon\). We study the quantitative homogenization for the solutions of the hyperbolic system \(\partial _t^2\mathbf{u}_\varepsilon =-B_\varepsilon\mathbf{u}_\varepsilon\). In operator terms, we are interested in approximations of the operators \(\cos (tB_\varepsilon ^{1/2})\) and \(B_\varepsilon ^{-1/2}\sin (tB_\varepsilon ^{1/2})\) in suitable operator norms. Approximations for the resolvent \(B_\varepsilon ^{-1}\) have been already obtained by T.A. Suslina. So, we rewrite hyperbolic equation as a system for the vector with components \(\mathbf{u}_\varepsilon \) and \(\partial _t\mathbf{u}_\varepsilon\), and consider the corresponding unitary group. For this group, we adapt the proof of the Trotter-Kato theorem by introduction of some correction term and derive hyperbolic results from elliptic ones.
期刊介绍：
Russian Journal of Mathematical Physics is a peer-reviewed periodical that deals with the full range of topics subsumed by that discipline, which lies at the foundation of much of contemporary science. Thus, in addition to mathematical physics per se, the journal coverage includes, but is not limited to, functional analysis, linear and nonlinear partial differential equations, algebras, quantization, quantum field theory, modern differential and algebraic geometry and topology, representations of Lie groups, calculus of variations, asymptotic methods, random process theory, dynamical systems, and control theory.
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