{"title":"粘性冲击与标量守恒律的长时间行为","authors":"Thierry Gallay, Arnd Scheel","doi":"10.3934/cpaa.2023119","DOIUrl":null,"url":null,"abstract":"We study the long-time behavior of scalar viscous conservation laws via the structure of $ \\omega $-limit sets. We show that $ \\omega $-limit sets always contain constants or shocks by establishing convergence to shocks for arbitrary monotone initial data. In the particular case of Burgers' equation, we review and refine results that parametrize entire solutions in terms of probability measures, and we construct initial data for which the $ \\omega $-limit set is not reduced to the translates of a single shock. Finally we propose several open problems related to the description of long-time dynamics.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"27 1","pages":"0"},"PeriodicalIF":1.0000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Viscous shocks and long-time behavior of scalar conservation laws\",\"authors\":\"Thierry Gallay, Arnd Scheel\",\"doi\":\"10.3934/cpaa.2023119\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the long-time behavior of scalar viscous conservation laws via the structure of $ \\\\omega $-limit sets. We show that $ \\\\omega $-limit sets always contain constants or shocks by establishing convergence to shocks for arbitrary monotone initial data. In the particular case of Burgers' equation, we review and refine results that parametrize entire solutions in terms of probability measures, and we construct initial data for which the $ \\\\omega $-limit set is not reduced to the translates of a single shock. Finally we propose several open problems related to the description of long-time dynamics.\",\"PeriodicalId\":10643,\"journal\":{\"name\":\"Communications on Pure and Applied Analysis\",\"volume\":\"27 1\",\"pages\":\"0\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications on Pure and Applied Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3934/cpaa.2023119\",\"RegionNum\":3,\"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 Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/cpaa.2023119","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Viscous shocks and long-time behavior of scalar conservation laws
We study the long-time behavior of scalar viscous conservation laws via the structure of $ \omega $-limit sets. We show that $ \omega $-limit sets always contain constants or shocks by establishing convergence to shocks for arbitrary monotone initial data. In the particular case of Burgers' equation, we review and refine results that parametrize entire solutions in terms of probability measures, and we construct initial data for which the $ \omega $-limit set is not reduced to the translates of a single shock. Finally we propose several open problems related to the description of long-time dynamics.
期刊介绍:
CPAA publishes original research papers of the highest quality in all the major areas of analysis and its applications, with a central theme on theoretical and numeric differential equations. Invited expository articles are also published from time to time. It is edited by a group of energetic leaders to guarantee the journal''s highest standard and closest link to the scientific communities.