一期Muskat问题的全局适定性

IF 3.1 1区数学 Q1 MATHEMATICS

Communications on Pure and Applied Mathematics Pub Date : 2023-07-11 DOI:10.1002/cpa.22124

Hongjie Dong, Francisco Gancedo, Huy Q. Nguyen

{"title":"一期Muskat问题的全局适定性","authors":"Hongjie Dong, Francisco Gancedo, Huy Q. Nguyen","doi":"10.1002/cpa.22124","DOIUrl":null,"url":null,"abstract":"<p>The free boundary problem for a two-dimensional fluid permeating a porous medium is studied. This is known as the one-phase Muskat problem and is mathematically equivalent to the vertical Hele-Shaw problem driven by gravity force. We prove that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global-in-time Lipschitz solution in the strong <math>\n <semantics>\n <mrow>\n <msubsup>\n <mi>L</mi>\n <mi>t</mi>\n <mi>∞</mi>\n </msubsup>\n <msubsup>\n <mi>L</mi>\n <mi>x</mi>\n <mn>2</mn>\n </msubsup>\n </mrow>\n <annotation>$L^\\infty _t L^2_x$</annotation>\n </semantics></math> sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.1000,"publicationDate":"2023-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"11","resultStr":"{\"title\":\"Global well-posedness for the one-phase Muskat problem\",\"authors\":\"Hongjie Dong, Francisco Gancedo, Huy Q. Nguyen\",\"doi\":\"10.1002/cpa.22124\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The free boundary problem for a two-dimensional fluid permeating a porous medium is studied. This is known as the one-phase Muskat problem and is mathematically equivalent to the vertical Hele-Shaw problem driven by gravity force. We prove that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global-in-time Lipschitz solution in the strong <math>\\n <semantics>\\n <mrow>\\n <msubsup>\\n <mi>L</mi>\\n <mi>t</mi>\\n <mi>∞</mi>\\n </msubsup>\\n <msubsup>\\n <mi>L</mi>\\n <mi>x</mi>\\n <mn>2</mn>\\n </msubsup>\\n </mrow>\\n <annotation>$L^\\\\infty _t L^2_x$</annotation>\\n </semantics></math> sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size.</p>\",\"PeriodicalId\":10601,\"journal\":{\"name\":\"Communications on Pure and Applied Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":3.1000,\"publicationDate\":\"2023-07-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"11\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications on Pure and Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22124\",\"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":"Communications on Pure and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22124","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 11

摘要

研究了二维流体在多孔介质中的自由边界问题。这就是众所周知的单相马斯喀特问题，在数学上等同于重力驱动下的垂直Hele - Shaw问题。我们证明了如果初始自由边界是周期Lipschitz函数的图，那么存在一个强Lt∞Lx2 $L^\infty _t L^2_x$意义上的全局时间Lipschitz解，并且它是唯一的黏性解。证明需要对层势和点向椭圆正则性进行定量估计。本文首次构造了具有任意大小初始数据的Muskat问题的唯一全局强解。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Global well-posedness for the one-phase Muskat problem

The free boundary problem for a two-dimensional fluid permeating a porous medium is studied. This is known as the one-phase Muskat problem and is mathematically equivalent to the vertical Hele-Shaw problem driven by gravity force. We prove that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global-in-time Lipschitz solution in the strong $L_{t}^{\infty} L_{x}^{2}$ sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Communications on Pure and Applied Mathematics 数学-数学

CiteScore

6.70

自引率

3.30%

发文量

审稿时长

>12 weeks

期刊介绍： Communications on Pure and Applied Mathematics (ISSN 0010-3640) is published monthly, one volume per year, by John Wiley & Sons, Inc. © 2019. The journal primarily publishes papers originating at or solicited by the Courant Institute of Mathematical Sciences. It features recent developments in applied mathematics, mathematical physics, and mathematical analysis. The topics include partial differential equations, computer science, and applied mathematics. CPAM is devoted to mathematical contributions to the sciences; both theoretical and applied papers, of original or expository type, are included.