{"title":"具有复合波的一维可压缩纳维-斯托克斯/阿伦-卡恩系统的锐界面极限","authors":"Ya-zhou Chen, Yi Peng, Xiao-ding Shi","doi":"10.1007/s10255-024-1130-7","DOIUrl":null,"url":null,"abstract":"<p>This paper is concerned with the sharp interface limit of Cauchy problem for the one-dimensional compressible Navier-Stokes/Allen-Cahn system with a composite wave consisting of the superposition of a rarefaction wave and a shock wave. Under the assumption that the viscosity coefficient and the reciprocal of mobility coefficient are directly proportional to the interface thickness, we first convert the sharp interface limit of the system into the large time behavior of the composite wave via a natural scaling. Then we prove that the composite wave is asymptotically stable under the small initial perturbations and the small strength of the rarefaction and shock wave. Finally, we show the solution of the Cauchy problem exists for all time, and converges to the composite wave solution of the corresponding Euler equations as the thickness of the interface tends to zero. The proof is mainly based on the energy method and the relative entropy.</p>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":"51 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Sharp Interface Limit for the One-dimensional Compressible Navier-Stokes/Allen-Cahn System with Composite Waves\",\"authors\":\"Ya-zhou Chen, Yi Peng, Xiao-ding Shi\",\"doi\":\"10.1007/s10255-024-1130-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>This paper is concerned with the sharp interface limit of Cauchy problem for the one-dimensional compressible Navier-Stokes/Allen-Cahn system with a composite wave consisting of the superposition of a rarefaction wave and a shock wave. Under the assumption that the viscosity coefficient and the reciprocal of mobility coefficient are directly proportional to the interface thickness, we first convert the sharp interface limit of the system into the large time behavior of the composite wave via a natural scaling. Then we prove that the composite wave is asymptotically stable under the small initial perturbations and the small strength of the rarefaction and shock wave. Finally, we show the solution of the Cauchy problem exists for all time, and converges to the composite wave solution of the corresponding Euler equations as the thickness of the interface tends to zero. The proof is mainly based on the energy method and the relative entropy.</p>\",\"PeriodicalId\":6951,\"journal\":{\"name\":\"Acta Mathematicae Applicatae Sinica, English Series\",\"volume\":\"51 1\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematicae Applicatae Sinica, English Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10255-024-1130-7\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematicae Applicatae Sinica, English Series","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10255-024-1130-7","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Sharp Interface Limit for the One-dimensional Compressible Navier-Stokes/Allen-Cahn System with Composite Waves
This paper is concerned with the sharp interface limit of Cauchy problem for the one-dimensional compressible Navier-Stokes/Allen-Cahn system with a composite wave consisting of the superposition of a rarefaction wave and a shock wave. Under the assumption that the viscosity coefficient and the reciprocal of mobility coefficient are directly proportional to the interface thickness, we first convert the sharp interface limit of the system into the large time behavior of the composite wave via a natural scaling. Then we prove that the composite wave is asymptotically stable under the small initial perturbations and the small strength of the rarefaction and shock wave. Finally, we show the solution of the Cauchy problem exists for all time, and converges to the composite wave solution of the corresponding Euler equations as the thickness of the interface tends to zero. The proof is mainly based on the energy method and the relative entropy.
期刊介绍:
Acta Mathematicae Applicatae Sinica (English Series) is a quarterly journal established by the Chinese Mathematical Society. The journal publishes high quality research papers from all branches of applied mathematics, and particularly welcomes those from partial differential equations, computational mathematics, applied probability, mathematical finance, statistics, dynamical systems, optimization and management science.