{"title":"Oscillations in Wave Map Systems and Homogenization of the Einstein Equations in Symmetry","authors":"André Guerra, Rita Teixeira da Costa","doi":"10.1007/s00205-024-02042-3","DOIUrl":null,"url":null,"abstract":"<div><p>In 1989, Burnett conjectured that, under appropriate assumptions, the limit of highly oscillatory solutions to the Einstein vacuum equations is a solution of the Einstein–massless Vlasov system. In a recent breakthrough, Huneau–Luk (Ann Sci l’ENS, 2024) gave a proof of the conjecture in <i>U</i>(1)-symmetry and elliptic gauge. They also require control on up to fourth order derivatives of the metric components. In this paper, we give a streamlined proof of a stronger result and, in the spirit of Burnett’s original conjecture, we remove the need for control on higher derivatives. Our methods also apply to general wave map equations.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2024-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02042-3.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Rational Mechanics and Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00205-024-02042-3","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In 1989, Burnett conjectured that, under appropriate assumptions, the limit of highly oscillatory solutions to the Einstein vacuum equations is a solution of the Einstein–massless Vlasov system. In a recent breakthrough, Huneau–Luk (Ann Sci l’ENS, 2024) gave a proof of the conjecture in U(1)-symmetry and elliptic gauge. They also require control on up to fourth order derivatives of the metric components. In this paper, we give a streamlined proof of a stronger result and, in the spirit of Burnett’s original conjecture, we remove the need for control on higher derivatives. Our methods also apply to general wave map equations.
期刊介绍：
The Archive for Rational Mechanics and Analysis nourishes the discipline of mechanics as a deductive, mathematical science in the classical tradition and promotes analysis, particularly in the context of application. Its purpose is to give rapid and full publication to research of exceptional moment, depth and permanence.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
