{"title":"具有球对称的相对论欧拉系统的自相似解","authors":"Bing-Ze Lu, Chou Kao, Wen-Ching Lien","doi":"10.1090/qam/1680","DOIUrl":null,"url":null,"abstract":"We consider the spherical piston problem in relativistic fluid dynamics. When the spherical piston expands at a constant speed, we show that the self-similar solution with a shock front exists under the relativistic principle that all velocities are bounded by the light speed. The equation of state is given by <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P equals sigma squared rho\"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>σ<!-- σ --></mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>ρ<!-- ρ --></mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">P= \\sigma ^2 \\rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sigma\"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">\\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the sound speed, is a constant. It is an important model describing the evolution of stars. Also, we present the global existence of BV solutions for the relativistic Euler system given that the piston speed is perturbed around a constant in a finite time interval. The analysis is based on the modified Glimm scheme and the smallness assumption is required on the initial data.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":"15 3","pages":"0"},"PeriodicalIF":0.9000,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Self-similar solutions of the relativistic Euler system with spherical symmetry\",\"authors\":\"Bing-Ze Lu, Chou Kao, Wen-Ching Lien\",\"doi\":\"10.1090/qam/1680\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider the spherical piston problem in relativistic fluid dynamics. When the spherical piston expands at a constant speed, we show that the self-similar solution with a shock front exists under the relativistic principle that all velocities are bounded by the light speed. The equation of state is given by <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper P equals sigma squared rho\\\"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>σ<!-- σ --></mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>ρ<!-- ρ --></mml:mi> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">P= \\\\sigma ^2 \\\\rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"sigma\\\"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the sound speed, is a constant. It is an important model describing the evolution of stars. Also, we present the global existence of BV solutions for the relativistic Euler system given that the piston speed is perturbed around a constant in a finite time interval. The analysis is based on the modified Glimm scheme and the smallness assumption is required on the initial data.\",\"PeriodicalId\":20964,\"journal\":{\"name\":\"Quarterly of Applied Mathematics\",\"volume\":\"15 3\",\"pages\":\"0\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-10-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Quarterly of Applied Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/qam/1680\",\"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":"Quarterly of Applied Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/qam/1680","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Self-similar solutions of the relativistic Euler system with spherical symmetry
We consider the spherical piston problem in relativistic fluid dynamics. When the spherical piston expands at a constant speed, we show that the self-similar solution with a shock front exists under the relativistic principle that all velocities are bounded by the light speed. The equation of state is given by P=σ2ρP= \sigma ^2 \rho, where σ\sigma, the sound speed, is a constant. It is an important model describing the evolution of stars. Also, we present the global existence of BV solutions for the relativistic Euler system given that the piston speed is perturbed around a constant in a finite time interval. The analysis is based on the modified Glimm scheme and the smallness assumption is required on the initial data.
期刊介绍：
The Quarterly of Applied Mathematics contains original papers in applied mathematics which have a close connection with applications. An author index appears in the last issue of each volume.
This journal, published quarterly by Brown University with articles electronically published individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website.
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