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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 g.v 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