{"title":"正则网络曲率运动的存在唯一性","authors":"Michael Gosswein, Julia Menzel, Alessandra Pluda","doi":"10.4171/ifb/477","DOIUrl":null,"url":null,"abstract":"We prove existence and uniqueness of the motion by curvatureof networks in $\\mathbb{R}^n$ when the initial datum is of class $W^{2-\\frac{2}{p}}_p$, with triple junction where the unit tangent vectors to the concurring curves form angles of $120$ degrees. Moreover we investigated the regularization effect due to the parabolic nature of the system. An application of this wellposedness result is a new proof of Theorem 3.18 in \"Motion by Curvature of Planar Networks\" by Mantegazza-Novaga-Tortorelli where the possible behaviors of the solutions at the maximal time of existence are described. Our study is motivated by an open question proposed in \"Evolution of Networks with Multiple Junctions \" by Mantegazza-Novaga-Pluda-Schulze: does there exist a unique solution of the motion by curvature of networks with initial datum a regular network of class $C^2$? We give a positive answer.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"50 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2020-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"10","resultStr":"{\"title\":\"Existence and uniqueness of the motion by curvature of regular networks\",\"authors\":\"Michael Gosswein, Julia Menzel, Alessandra Pluda\",\"doi\":\"10.4171/ifb/477\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove existence and uniqueness of the motion by curvatureof networks in $\\\\mathbb{R}^n$ when the initial datum is of class $W^{2-\\\\frac{2}{p}}_p$, with triple junction where the unit tangent vectors to the concurring curves form angles of $120$ degrees. Moreover we investigated the regularization effect due to the parabolic nature of the system. An application of this wellposedness result is a new proof of Theorem 3.18 in \\\"Motion by Curvature of Planar Networks\\\" by Mantegazza-Novaga-Tortorelli where the possible behaviors of the solutions at the maximal time of existence are described. Our study is motivated by an open question proposed in \\\"Evolution of Networks with Multiple Junctions \\\" by Mantegazza-Novaga-Pluda-Schulze: does there exist a unique solution of the motion by curvature of networks with initial datum a regular network of class $C^2$? We give a positive answer.\",\"PeriodicalId\":13863,\"journal\":{\"name\":\"Interfaces and Free Boundaries\",\"volume\":\"50 1\",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2020-03-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"10\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Interfaces and Free Boundaries\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4171/ifb/477\",\"RegionNum\":4,\"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":"Interfaces and Free Boundaries","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/ifb/477","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Existence and uniqueness of the motion by curvature of regular networks
We prove existence and uniqueness of the motion by curvatureof networks in $\mathbb{R}^n$ when the initial datum is of class $W^{2-\frac{2}{p}}_p$, with triple junction where the unit tangent vectors to the concurring curves form angles of $120$ degrees. Moreover we investigated the regularization effect due to the parabolic nature of the system. An application of this wellposedness result is a new proof of Theorem 3.18 in "Motion by Curvature of Planar Networks" by Mantegazza-Novaga-Tortorelli where the possible behaviors of the solutions at the maximal time of existence are described. Our study is motivated by an open question proposed in "Evolution of Networks with Multiple Junctions " by Mantegazza-Novaga-Pluda-Schulze: does there exist a unique solution of the motion by curvature of networks with initial datum a regular network of class $C^2$? We give a positive answer.
期刊介绍:
Interfaces and Free Boundaries is dedicated to the mathematical modelling, analysis and computation of interfaces and free boundary problems in all areas where such phenomena are pertinent. The journal aims to be a forum where mathematical analysis, partial differential equations, modelling, scientific computing and the various applications which involve mathematical modelling meet. Submissions should, ideally, emphasize the combination of theory and application.