tailieunhanh - Đề tài " Rough solutions of the Einstein-vacuum equations "

This is the first in a series of 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. Following [Kl-Ro] we develop new analytic methods based on Strichartz-type inequalities which result in a gain of half a derivative relative to the classical result. . | Annals of Mathematics Rough solutions of the Einstein-vacuum equations By Sergiu Klainerman and Igor Rodnianski Annals of Mathematics 161 2005 1143 1193 Rough solutions of the Einstein-vacuum equations By Sergiu Klainerman and Igor Rodnianski To Y. Choquet-Bruhat in honour of the 50th anniversary of her fundamental paper Br on the Cauchy problem in General Relativity Abstract This is the first in a series of 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. Following Kl-Ro we develop new analytic methods based on Strichartz-type inequalities which result in a gain of half a derivative relative to the classical result. Our methods blend paradifferential techniques with a geometric approach to the derivation of decay estimates. The latter allows us to take full advantage of the specific structure of the Einstein equations. 1. Introduction We consider the Einstein-vacuum equations 1 Rafi g 0 where g is a four-dimensional Lorentz metric and Rafi its Ricci curvature tensor. In wave coordinates xa 2 Ogxa ill d g v g ổv xa 0 g the Einstein-vacuum equations take the reduced form see Br H-K-M . 3 gafi dadfi Nfv g dg with N quadratic in the first derivatives dg of the metric. We consider the initial value problem along the spacelike hyperplane s given by t x0 0 4 Vgafi 0 e Hs-1 s dtgafi 0 e Hs-1 s 1144 SERGIU KLAINERMAN AND IGOR RODNIANSKI with V denoting the gradient with respect to the space coordinates xi i 1 2 3 and Hs the standard Sobolev spaces. We also assume that gaj 0 is a continuous Lorentz metric and 5 sup go 0 - maạI 0 as r x x r where x Q23 1 x 2 2 and map is the Minkowski metric. The following local existence and uniqueness result well-posedness is well known see H-K-M and the previous result of Ch. Bruhat Br for

• Thạc sĩ - Tiến sĩ - Cao học
• Rough solutions
• subjects of mathematics
• science reports
• studied mathematics
• natural sciences
• scientific research
• Journal of Chemistry
• Boron based materials
• Direct borohydride fuel cell
• Hydrogen storage
• Sodium borohydride
• Aqueous solutions
• 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