Type-0 singularities in the network flow – Evolution of trees

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2022-07-08 DOI:10.1515/crelle-2022-0055

C. Mantegazza, M. Novaga, Alessandra Pluda

{"title":"Type-0 singularities in the network flow – Evolution of trees","authors":"C. Mantegazza, M. Novaga, Alessandra Pluda","doi":"10.1515/crelle-2022-0055","DOIUrl":null,"url":null,"abstract":"Abstract The motion by curvature of networks is the generalization to finite union of curves of the curve shortening flow. This evolution has several peculiar features, mainly due to the presence of junctions where the curves meet. In this paper we show that whenever the length of one single curve vanishes and two triple junctions coalesce, then the curvature of the evolving networks remains bounded. This topological singularity is exclusive of the network flow and it can be referred as a “Type-0” singularity, in contrast to the well known “Type-I” and “Type-II” ones of the usual mean curvature flow of smooth curves or hypersurfaces, characterized by the different rates of blow-up of the curvature. As a consequence, we are able to give a complete description of the evolution of tree–like networks till the first singular time, under the assumption that all the “tangents flows” have unit multiplicity. If the lifespan of such solutions is finite, then the curvature of the network remains bounded and we can apply the results from [T. Ilmanen, A. Neves and F. Schulze, On short time existence for the planar network flow, J. Differential Geom. 111 2019, 1, 39–89] and [J. Lira, R. Mazzeo, A. Pluda and M. Saez, Short–time existence for the network flow, preprint 2021] to “restart” the flow after the singularity.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2022-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/crelle-2022-0055","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}

引用次数: 3

Abstract

Abstract The motion by curvature of networks is the generalization to finite union of curves of the curve shortening flow. This evolution has several peculiar features, mainly due to the presence of junctions where the curves meet. In this paper we show that whenever the length of one single curve vanishes and two triple junctions coalesce, then the curvature of the evolving networks remains bounded. This topological singularity is exclusive of the network flow and it can be referred as a “Type-0” singularity, in contrast to the well known “Type-I” and “Type-II” ones of the usual mean curvature flow of smooth curves or hypersurfaces, characterized by the different rates of blow-up of the curvature. As a consequence, we are able to give a complete description of the evolution of tree–like networks till the first singular time, under the assumption that all the “tangents flows” have unit multiplicity. If the lifespan of such solutions is finite, then the curvature of the network remains bounded and we can apply the results from [T. Ilmanen, A. Neves and F. Schulze, On short time existence for the planar network flow, J. Differential Geom. 111 2019, 1, 39–89] and [J. Lira, R. Mazzeo, A. Pluda and M. Saez, Short–time existence for the network flow, preprint 2021] to “restart” the flow after the singularity.

查看原文本刊更多论文

网络流中的0型奇点——树的演化

网络的曲率运动是对曲线缩短流的曲线有限并的推广。这种演变有几个特殊的特征，主要是由于曲线相交处存在连接点。在本文中，我们证明了当一条单曲线的长度消失并且两个三重结点合并时，进化网络的曲率保持有界。这种拓扑奇点不包含网络流，它可以被称为“0型”奇点，而不是众所周知的光滑曲线或超曲面的通常平均曲率流的“i型”和“ii型”奇点，其特征是曲率的膨胀率不同。因此，在假设所有“切线流”具有单位多重性的前提下，我们能够给出树状网络直到第一个奇异时间的进化的完整描述。如果这些解的寿命是有限的，那么网络的曲率仍然是有界的，我们可以应用[T]的结果。张建军，张建军，张建军，等。平面网络流的短时间存在性研究[J] .地球物理学报，2019，(1):39 - 39。Lira, R. Mazzeo, A. Pluda和M. Saez，网络流的短时存在，预印本2021]。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.