{"title":"关于马斯卡特方程的柯西问题。II： 关键初始数据","authors":"Thomas Alazard, Quoc-Hung Nguyen","doi":"10.1007/s40818-021-00099-x","DOIUrl":null,"url":null,"abstract":"<div><p>We prove that the Cauchy problem for the Muskat equation is well-posed locally in time for any initial data in the critical space of Lipschitz functions with three-half derivative in <span>\\(L^2\\)</span>. Moreover, we prove that the solution exists globally in time under a smallness assumption.</p></div>","PeriodicalId":36382,"journal":{"name":"Annals of Pde","volume":"7 1","pages":""},"PeriodicalIF":2.4000,"publicationDate":"2021-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40818-021-00099-x","citationCount":"1","resultStr":"{\"title\":\"On the Cauchy Problem for the Muskat Equation. II: Critical Initial Data\",\"authors\":\"Thomas Alazard, Quoc-Hung Nguyen\",\"doi\":\"10.1007/s40818-021-00099-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We prove that the Cauchy problem for the Muskat equation is well-posed locally in time for any initial data in the critical space of Lipschitz functions with three-half derivative in <span>\\\\(L^2\\\\)</span>. Moreover, we prove that the solution exists globally in time under a smallness assumption.</p></div>\",\"PeriodicalId\":36382,\"journal\":{\"name\":\"Annals of Pde\",\"volume\":\"7 1\",\"pages\":\"\"},\"PeriodicalIF\":2.4000,\"publicationDate\":\"2021-04-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1007/s40818-021-00099-x\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annals of Pde\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40818-021-00099-x\",\"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/s40818-021-00099-x","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
On the Cauchy Problem for the Muskat Equation. II: Critical Initial Data
We prove that the Cauchy problem for the Muskat equation is well-posed locally in time for any initial data in the critical space of Lipschitz functions with three-half derivative in \(L^2\). Moreover, we prove that the solution exists globally in time under a smallness assumption.
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