{"title":"相变动力学中分裂合并模式的数学理论","authors":"Eva Kardhashi, M. Laforest, P. LeFloch","doi":"10.1080/03605302.2022.2053862","DOIUrl":null,"url":null,"abstract":"Abstract For nonlinear hyperbolic systems of conservation laws in one space variable, we establish the existence of nonclassical entropy solutions exhibiting nonlinear interactions between shock waves with strong strength. The proposed theory is relevant in the theory of phase transition dynamics, and the solutions under consideration enjoy a splitting–merging pattern, consisting of (compressive) classical and (undercompressive) nonclassical waves, interacting together as well as with classical waves of weaker strength. Our analysis is based on three novel ideas. First, a generalization of Hayes–LeFloch’s nonclassical Riemann solver is introduced for systems and is based on prescribing, on one hand, a kinetic relation for the propagation of nonclassical undercompressive shocks and, on the other hand, a nucleation criterion that selects between classical and nonclassical behavior. Second, we extend LeFloch-Shearer’s theorem to systems and we prove that the presence of a nucleation condition implies that only a finite number of splitting and merging cycles can occur. Third, our arguments of nonlinear stability build upon recent work by the last two authors who identified a natural total variation functional for scalar conservation laws and, specifically, for systems of conservation laws we introduce here novel functionals which measure the total variation and wave interaction of nonclassical and classical waves.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"1339 - 1393"},"PeriodicalIF":2.1000,"publicationDate":"2021-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The mathematical theory of splitting-merging patterns in phase transition dynamics\",\"authors\":\"Eva Kardhashi, M. Laforest, P. LeFloch\",\"doi\":\"10.1080/03605302.2022.2053862\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract For nonlinear hyperbolic systems of conservation laws in one space variable, we establish the existence of nonclassical entropy solutions exhibiting nonlinear interactions between shock waves with strong strength. The proposed theory is relevant in the theory of phase transition dynamics, and the solutions under consideration enjoy a splitting–merging pattern, consisting of (compressive) classical and (undercompressive) nonclassical waves, interacting together as well as with classical waves of weaker strength. Our analysis is based on three novel ideas. First, a generalization of Hayes–LeFloch’s nonclassical Riemann solver is introduced for systems and is based on prescribing, on one hand, a kinetic relation for the propagation of nonclassical undercompressive shocks and, on the other hand, a nucleation criterion that selects between classical and nonclassical behavior. Second, we extend LeFloch-Shearer’s theorem to systems and we prove that the presence of a nucleation condition implies that only a finite number of splitting and merging cycles can occur. Third, our arguments of nonlinear stability build upon recent work by the last two authors who identified a natural total variation functional for scalar conservation laws and, specifically, for systems of conservation laws we introduce here novel functionals which measure the total variation and wave interaction of nonclassical and classical waves.\",\"PeriodicalId\":50657,\"journal\":{\"name\":\"Communications in Partial Differential Equations\",\"volume\":\"47 1\",\"pages\":\"1339 - 1393\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2021-05-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in Partial Differential Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1080/03605302.2022.2053862\",\"RegionNum\":2,\"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 in Partial Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1080/03605302.2022.2053862","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
The mathematical theory of splitting-merging patterns in phase transition dynamics
Abstract For nonlinear hyperbolic systems of conservation laws in one space variable, we establish the existence of nonclassical entropy solutions exhibiting nonlinear interactions between shock waves with strong strength. The proposed theory is relevant in the theory of phase transition dynamics, and the solutions under consideration enjoy a splitting–merging pattern, consisting of (compressive) classical and (undercompressive) nonclassical waves, interacting together as well as with classical waves of weaker strength. Our analysis is based on three novel ideas. First, a generalization of Hayes–LeFloch’s nonclassical Riemann solver is introduced for systems and is based on prescribing, on one hand, a kinetic relation for the propagation of nonclassical undercompressive shocks and, on the other hand, a nucleation criterion that selects between classical and nonclassical behavior. Second, we extend LeFloch-Shearer’s theorem to systems and we prove that the presence of a nucleation condition implies that only a finite number of splitting and merging cycles can occur. Third, our arguments of nonlinear stability build upon recent work by the last two authors who identified a natural total variation functional for scalar conservation laws and, specifically, for systems of conservation laws we introduce here novel functionals which measure the total variation and wave interaction of nonclassical and classical waves.
期刊介绍:
This journal aims to publish high quality papers concerning any theoretical aspect of partial differential equations, as well as its applications to other areas of mathematics. Suitability of any paper is at the discretion of the editors. We seek to present the most significant advances in this central field to a wide readership which includes researchers and graduate students in mathematics and the more mathematical aspects of physics and engineering.