{"title":"6 Decay Estimates for and (Theorems M2, M3)","authors":"","doi":"10.1515/9780691218526-008","DOIUrl":"https://doi.org/10.1515/9780691218526-008","url":null,"abstract":"","PeriodicalId":371134,"journal":{"name":"Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125141875","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}
{"title":"10 Regge-Wheeler Type Equations","authors":"","doi":"10.1515/9780691218526-012","DOIUrl":"https://doi.org/10.1515/9780691218526-012","url":null,"abstract":"","PeriodicalId":371134,"journal":{"name":"Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126940185","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}
{"title":"List of Figures","authors":"","doi":"10.1515/9780691218526-001","DOIUrl":"https://doi.org/10.1515/9780691218526-001","url":null,"abstract":"","PeriodicalId":371134,"journal":{"name":"Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122081016","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}
{"title":"B Appendix to Chapter 8","authors":"","doi":"10.1515/9780691218526-014","DOIUrl":"https://doi.org/10.1515/9780691218526-014","url":null,"abstract":"","PeriodicalId":371134,"journal":{"name":"Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations","volume":"137 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127013937","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}
{"title":"C Appendix to Chapter 9","authors":"","doi":"10.1515/9780691218526-015","DOIUrl":"https://doi.org/10.1515/9780691218526-015","url":null,"abstract":"","PeriodicalId":371134,"journal":{"name":"Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121827617","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}
{"title":"List of Figures","authors":"","doi":"10.2307/j.ctv15r57cw.3","DOIUrl":"https://doi.org/10.2307/j.ctv15r57cw.3","url":null,"abstract":"","PeriodicalId":371134,"journal":{"name":"Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations","volume":"46 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":"122545413","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}
{"title":"Decay Estimates for q (Theorem M1)","authors":"S. Klainerman, J. Szeftel","doi":"10.23943/PRINCETON/9780691212425.003.0005","DOIUrl":"https://doi.org/10.23943/PRINCETON/9780691212425.003.0005","url":null,"abstract":"This chapter examines the proof for Theorem M1, deriving decay estimates for the quantity q for k ≤ k\u0000 small + 20 derivatives. To this end, it uses the wave equation satisfied by q. The spacetime M is decomposed as M = (int)M u (ext)M and that u is an outgoing optical function on (ext)M while u is an ingoing optical function. The chapter relies on the global frame defined in section 3.5, and r and m denote the corresponding scalar functions associated to it. It also proves two theorems on improved weighted estimates, as well as flux decay estimates for q.","PeriodicalId":371134,"journal":{"name":"Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations","volume":"2003 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":"127323752","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}
{"title":"GCM Procedure","authors":"S. Klainerman, J. Szeftel","doi":"10.2307/j.ctv15r57cw.13","DOIUrl":"https://doi.org/10.2307/j.ctv15r57cw.13","url":null,"abstract":"This chapter describes the general covariant modulation (GCM) procedure in detail. It considers an axially symmetric polarized spacetime region R foliated by two functions (u, s) such that: on R, (u, s) defines an outgoing geodesic foliation as in section 2.2.4. The chapter then outlines the elliptic Hodge lemma. It also looks at the deformations of S surfaces, frame transformations, and the existence of GCM spheres. It recalls the transformation formulas recorded in Proposition 2.90, before rewriting a subset of these transformations in a more useful form. In the proof of existence and uniqueness of GCMS, one needs, in addition to the equations derived so far, an equation for the average of α. Finally, the chapter discusses the construction of GCM hypersurfaces.","PeriodicalId":371134,"journal":{"name":"Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations","volume":"24 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":"122016410","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}