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Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations最新文献

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Initialization and Extension (Theorems M6, M7, M8) 初始化与扩展(定理M6, M7, M8)
Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations Pub Date : 2020-12-15 DOI: 10.2307/j.ctv15r57cw.12
S. Klainerman, J. Szeftel
{"title":"Initialization and Extension (Theorems M6, M7, M8)","authors":"S. Klainerman, J. Szeftel","doi":"10.2307/j.ctv15r57cw.12","DOIUrl":"https://doi.org/10.2307/j.ctv15r57cw.12","url":null,"abstract":"This chapter focuses on the proof for Theorem M6 concerning initialization, Theorem M7 concerning extension, and Theorem M8 concerning the improvement of higher order weighted energies. It first improves the bootstrap assumptions on decay estimates. The chapter then improves the bootstrap assumptions on energies and weighted energies for R and Γ‎ relying on an iterative procedure which recovers derivatives one by one. It also outlines the norms for measuring weighted energies for curvature components and Ricci coefficients. To prove Theorem M8, the chapter relies on Propositions 8.11, 8.12, and 8.13. Among these propositions, only the last two involve the dangerous boundary term.","PeriodicalId":371134,"journal":{"name":"Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116529375","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Decay Estimates for 𝛼 and $underline{alpha }$ (Theorems M2, M3) 估计的衰减 $underline{alpha }$ (定理M2, M3)
Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations Pub Date : 2020-12-15 DOI: 10.2307/j.ctv15r57cw.10
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引用次数: 0
Decay Estimates for $mathfrak{q}$ (Theorem M1) $mathfrak{q}$的衰减估计(定理M1)
Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations Pub Date : 2020-12-15 DOI: 10.2307/j.ctv15r57cw.9
{"title":"Decay Estimates for $mathfrak{q}$ (Theorem M1)","authors":"","doi":"10.2307/j.ctv15r57cw.9","DOIUrl":"https://doi.org/10.2307/j.ctv15r57cw.9","url":null,"abstract":"","PeriodicalId":371134,"journal":{"name":"Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126303319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Decay Estimates (Theorems M4, M5) 衰减估计(定理M4, M5)
Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations Pub Date : 2020-12-15 DOI: 10.2307/j.ctv15r57cw.11
S. Klainerman, J. Szeftel
{"title":"Decay Estimates (Theorems M4, M5)","authors":"S. Klainerman, J. Szeftel","doi":"10.2307/j.ctv15r57cw.11","DOIUrl":"https://doi.org/10.2307/j.ctv15r57cw.11","url":null,"abstract":"This chapter evaluates the proof for Theorems M4 and M5. It relies on the decay of q, α‎ and α‎ to prove the decay estimates for all the other quantities. More precisely, the chapter relies on the results of Theorems M1, M2, and M3 to prove Theorems M4 and M5. The detailed proof of Theorem M4 provides the main decay estimates in (ext)M. The proof depends in a fundamental way on the geometric properties of the GCM hypersurface Σ‎∗, the spacelike future boundary of (ext)M introduced in section 3.1.2. The chapter then reformulates the main bootstrap assumptions in the form needed in the proof of Theorem M4.","PeriodicalId":371134,"journal":{"name":"Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123320999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Regge-Wheeler Type Equations Regge-Wheeler型方程
Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations Pub Date : 2020-12-15 DOI: 10.2307/j.ctv15r57cw.14
S. Klainerman, J. Szeftel
{"title":"Regge-Wheeler Type Equations","authors":"S. Klainerman, J. Szeftel","doi":"10.2307/j.ctv15r57cw.14","DOIUrl":"https://doi.org/10.2307/j.ctv15r57cw.14","url":null,"abstract":"This chapter explores estimates for Regge-Wheeler type wave equations used in Theorem M1. It first proves basic Morawetz estimates for ψ‎. The chapter then proves rp-weighted estimates in the spirit of Dafermos and Rodnianski for ψ‎. In particular, it obtains as an immediate corollary the proof of Theorem 5.17 in the case s = 0 (i.e., without commutating the equation of ψ‎ with derivatives). It also uses a variation of the method of [5] to derive slightly stronger weighted estimates and prove Theorem 5.18 in the case s = 0. Finally, commuting the equation of ψ‎ with derivatives, the chapter completes the proof of Theorem 5.17 by controlling higher order derivatives of ψ‎.","PeriodicalId":371134,"journal":{"name":"Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133350070","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Main Theorem 主要定理
Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations Pub Date : 2020-12-15 DOI: 10.1007/BFB0061169
C. J. Mozzochi
{"title":"Main Theorem","authors":"C. J. Mozzochi","doi":"10.1007/BFB0061169","DOIUrl":"https://doi.org/10.1007/BFB0061169","url":null,"abstract":"","PeriodicalId":371134,"journal":{"name":"Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127421076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Acknowledgments 致谢
Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations Pub Date : 2020-12-15 DOI: 10.2307/j.ctv15r57cw.4
{"title":"Acknowledgments","authors":"","doi":"10.2307/j.ctv15r57cw.4","DOIUrl":"https://doi.org/10.2307/j.ctv15r57cw.4","url":null,"abstract":"","PeriodicalId":371134,"journal":{"name":"Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations","volume":"242 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134071665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Main Theorem 主要定理
Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations Pub Date : 2020-12-15 DOI: 10.1016/s0079-8169(08)62338-7
S. Klainerman, J. Szeftel
{"title":"Main Theorem","authors":"S. Klainerman, J. Szeftel","doi":"10.1016/s0079-8169(08)62338-7","DOIUrl":"https://doi.org/10.1016/s0079-8169(08)62338-7","url":null,"abstract":"","PeriodicalId":371134,"journal":{"name":"Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations","volume":"490 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123194051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Preliminaries 预赛
Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations Pub Date : 2020-12-15 DOI: 10.23943/princeton/9780691212425.003.0002
S. Klainerman, J. Szeftel
{"title":"Preliminaries","authors":"S. Klainerman, J. Szeftel","doi":"10.23943/princeton/9780691212425.003.0002","DOIUrl":"https://doi.org/10.23943/princeton/9780691212425.003.0002","url":null,"abstract":"This chapter discusses the main quantities, equations, and basic tools needed in the following chapters. It is the main reference kit providing all main null structure and null Bianchi equations, in general null frames, in the context of axially symmetric polarized spacetimes. The chapter translates the null structure and null Bianchi identities associated to an S-foliation in the reduced picture. It starts with general, Z-invariant, S-foliation, before considering the special case of geodesic foliations. The chapter then looks at perturbations of Schwarzschild and invariant quantities. It investigates how the main Ricci and curvature quantities change relative to frame transformations, i.e., linear transformations which take null frames into null frames. Finally, the chapter presents wave equations for the invariant quantities.","PeriodicalId":371134,"journal":{"name":"Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125538701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Consequences of the Bootstrap Assumptions Bootstrap假设的结果
Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations Pub Date : 2020-12-15 DOI: 10.2307/j.ctv15r57cw.8
S. Klainerman, J. Szeftel
{"title":"Consequences of the Bootstrap Assumptions","authors":"S. Klainerman, J. Szeftel","doi":"10.2307/j.ctv15r57cw.8","DOIUrl":"https://doi.org/10.2307/j.ctv15r57cw.8","url":null,"abstract":"This chapter discusses the proof for Theorem M0, together with other first consequences of the bootstrap assumptions. The only bootstrap assumption used in the proof of Theorem M0 is the bootstrap assumption BA-D on decay for k = 0, 1 derivatives. The chapter then relies on (4.1.5) and the assumptions (4.1.1) on the initial data layer. This observation allows one to use the conclusions of Theorem M0, not only for the bootstrap spacetime M in Theorem M1–M5, but also for the extended spacetime in the proof of Theorem M8, where the only assumption is the one on decay (which is established for the extended spacetime in Theorem M7). The chapter not only improves the bootstrap assumption (4.1.7), but also gains derivatives iteratively.","PeriodicalId":371134,"journal":{"name":"Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129798230","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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