TAILIEUCHUNG - Đề tài " The causal structure of microlocalized rough Einstein metrics "

This is the second in a series of three papers in which we initiate the study of very rough solutions to the initial value problem for the Einstein vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and Sobolev inequalities. In this paper we develop the geometric analysis of the Eikonal equation for microlocalized rough Einstein metrics. This is a crucial step in the derivation of the decay estimates needed in the first paper. . | Annals of Mathematics The causal structure of microlocalized rough Einstein metrics By Sergiu Klainerman and Igor Rodnianski Annals of Mathematics 161 2005 1195 1243 The causal structure of microlocalized rough Einstein metrics By Sergiu Klainerman and Igor Rodnianski Abstract This is the second in a series of three papers in which we initiate the study of very rough solutions to the initial value problem for the Einstein vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and Sobolev inequalities. In this paper we develop the geometric analysis of the Eikonal equation for microlocalized rough Einstein metrics. This is a crucial step in the derivation of the decay estimates needed in the first paper. 1. Introduction This is the second in a series of three papers in which we initiate the study of very rough solutions of the Einstein vacuum equations. By very rough we mean solutions which cannot be dealt with by the classical techniques of energy estimates and Sobolev inequalities. In fact in this work we develop and take advantage of Strichartz-type estimates. The result stated in our first paper Kl-Ro1 is in fact optimal with respect to the full potential of such We recall below our main result Theorem Main Theorem . Let g be a classical solution2 of the Einstein equations 1 Rag g 0 expressed3 relative to wave coordinates xa 2 Dg xa A. d v g ỡv xa 0. g 1To go beyond our result will require the development of bilinear techniques for the Einstein equations see the discussion in the introduction to Kl-Ro1 . 2We denote by Rap the Ricci curvature of g. 3In wave coordinates the Einstein equations take the reduced form g JdadpgỊIv NỊtv g ỡg with N quadratic in the first derivatives dg of the metric. 1196 SERGIU KLAINERMAN AND IGOR RODNIANSKI Assume that on the initial spacelike hyperplane s given by t x0 0 V g. 0 e Hs-1 T dtgag 0 e Hs-1 T with V

• Thạc sĩ - Tiến sĩ - Cao học
• causal structure
• subjects of mathematics
• science reports
• studied mathematics
• natural sciences
• scientific research
• Genome Biology
• Peach genome assembly
• Structure variation
• Population dynamics
• Genomewide association study
• Large heterozygous inversion
• Argument Structure
• Counterconsiderations
• Causal Inductive Arguments
• Propositional Logic
• Relevance
• Inductive Arguments
• Research methods
• Data collection methods
• Understanding scientific research
• Ensuring research validity
• Measurement techniques
• Sampling methods
• Scientific research capacity
• Educational institutions
• Policy and management studies
• Vietnam Academy for Ethnic minorities
• Scientific bases
• earth sciences
• Research
• VNU Journal of Science
• scientific reports
• Action research
• linguistics
• research projects
• Foreign Languages
• cooperative research
• Renewable energy
• Introduction to research
• Scientific investigation
• Scientific investigation research
• Experimental designs
• Measurement of variables
• Reading research
• Conducting and reading research
• Research process
• Research report
• Purpose of research
• Scientific method
• Conducting research
• The effect of university autonomy on training and scientific research
• Vietnam national university of agriculture
• Training and scientific research
• Vascular plant
• Flora
• Yen Tu
• Quang Ninh Research Forest
• statements and documents scientific reports
• medical reports
• medical knowledge
• medical research
• clinical trials
• medical management
• health care
• Methods in behavioral research
• Behavioral research
• Scientific understanding
• Fundamental research issues
• Measurement concepts
• Observational methods
• reading comprehension
• education
• optimal picket
• model creation
• urban expansion
• organizing spatial
• Journal of Science
• Journal of Technology
• Technology Management
• Research quality
• Strategy and policy
• Summary of Doctoral thesis
• Summary of Doctoral thesis in public Management
• State management
• Female scientific research human
• Research human resource development
• Scientific research and teaching
• Defense and security education
• Current situation and solutions
• Security education
• Structural biochemistry
• epigenetics
• MicroRNAs
• Tạp chí khoa học
• Tạp chí Công nghệ
• Quản lý công nghệ
• Human resources
• Science management
• Poliomyelitis vaccine Research
• Non Ri Maprec
• Production center