{"title":"FLRW几何中自重力无旋转Chaplygin流体的非线性稳定性","authors":"Philippe G. LeFloch , Changhua Wei","doi":"10.1016/j.anihpc.2020.09.005","DOIUrl":null,"url":null,"abstract":"<div><p>We analyze the global nonlinear stability of FLRW (Friedmann-Lemaître-Robertson-Walker) spacetimes in the presence of an irrotational perfect fluid. We assume that the fluid is governed by the so-called (generalized) Chaplygin equation of state <span><math><mi>p</mi><mo>=</mo><mo>−</mo><mfrac><mrow><msup><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>ρ</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></math></span> relating the pressure to the mass-energy density, in which <span><math><mi>A</mi><mo>></mo><mn>0</mn></math></span> and <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span><span><span> are constants. We express the Einstein equations in wave gauge as a system of coupled nonlinear wave equations and, after performing a </span>conformal transformation, we analyze the global behavior of solutions toward the future. Under small perturbations, the </span><span><math><mo>(</mo><mn>3</mn><mo>+</mo><mn>1</mn><mo>)</mo></math></span><span>-spacetime metric, the mass-energy density, and the velocity vector describing the geometry and fluid unknowns remain globally close to a reference FLRW solution. Our analysis provides also the precise asymptotic behavior of the perturbed solutions toward the future.</span></p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":"38 3","pages":"Pages 787-814"},"PeriodicalIF":1.8000,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.09.005","citationCount":"13","resultStr":"{\"title\":\"Nonlinear stability of self-gravitating irrotational Chaplygin fluids in a FLRW geometry\",\"authors\":\"Philippe G. LeFloch , Changhua Wei\",\"doi\":\"10.1016/j.anihpc.2020.09.005\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We analyze the global nonlinear stability of FLRW (Friedmann-Lemaître-Robertson-Walker) spacetimes in the presence of an irrotational perfect fluid. We assume that the fluid is governed by the so-called (generalized) Chaplygin equation of state <span><math><mi>p</mi><mo>=</mo><mo>−</mo><mfrac><mrow><msup><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>ρ</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></math></span> relating the pressure to the mass-energy density, in which <span><math><mi>A</mi><mo>></mo><mn>0</mn></math></span> and <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span><span><span> are constants. We express the Einstein equations in wave gauge as a system of coupled nonlinear wave equations and, after performing a </span>conformal transformation, we analyze the global behavior of solutions toward the future. Under small perturbations, the </span><span><math><mo>(</mo><mn>3</mn><mo>+</mo><mn>1</mn><mo>)</mo></math></span><span>-spacetime metric, the mass-energy density, and the velocity vector describing the geometry and fluid unknowns remain globally close to a reference FLRW solution. Our analysis provides also the precise asymptotic behavior of the perturbed solutions toward the future.</span></p></div>\",\"PeriodicalId\":55514,\"journal\":{\"name\":\"Annales De L Institut Henri Poincare-Analyse Non Lineaire\",\"volume\":\"38 3\",\"pages\":\"Pages 787-814\"},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2021-05-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.09.005\",\"citationCount\":\"13\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annales De L Institut Henri Poincare-Analyse Non Lineaire\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0294144920300901\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0294144920300901","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Nonlinear stability of self-gravitating irrotational Chaplygin fluids in a FLRW geometry
We analyze the global nonlinear stability of FLRW (Friedmann-Lemaître-Robertson-Walker) spacetimes in the presence of an irrotational perfect fluid. We assume that the fluid is governed by the so-called (generalized) Chaplygin equation of state relating the pressure to the mass-energy density, in which and are constants. We express the Einstein equations in wave gauge as a system of coupled nonlinear wave equations and, after performing a conformal transformation, we analyze the global behavior of solutions toward the future. Under small perturbations, the -spacetime metric, the mass-energy density, and the velocity vector describing the geometry and fluid unknowns remain globally close to a reference FLRW solution. Our analysis provides also the precise asymptotic behavior of the perturbed solutions toward the future.
期刊介绍:
The Nonlinear Analysis section of the Annales de l''Institut Henri Poincaré is an international journal created in 1983 which publishes original and high quality research articles. It concentrates on all domains concerned with nonlinear analysis, specially applicable to PDE, mechanics, physics, economy, without overlooking the numerical aspects.