Regularity and geometric properties of solutions of the Einstein-Vacuum equations
Journées équations aux dérivées partielles (2002), article no. 15, 14 p.

We review recent results concerning the study of rough solutions to the initial value problem for the Einstein vacuum equations expressed relative to wave coordinates. We develop new analytic methods based on Strichartz type inequalities which results 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.

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     title = {Regularity and geometric properties of solutions of the {Einstein-Vacuum} equations},
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     publisher = {Universit\'e de Nantes},
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Klainerman, Sergiu; Rodnianski, Igor. Regularity and geometric properties of solutions of the Einstein-Vacuum equations. Journées équations aux dérivées partielles (2002), article  no. 15, 14 p. doi : 10.5802/jedp.613. http://www.numdam.org/articles/10.5802/jedp.613/

[An-Mo] L. Andersson and V. Moncrief Elliptic-hyperbolic systems and the Einstein equations. preprint | MR

[Ba-Ch1] H. Bahouri and J. Y. Chemin. Équations d'ondes quasilinéaires et estimation de Strichartz. Amer. J. Math., vol. 121; (1999), pp. 1337-1777 | MR | Zbl

[Ba-Ch2] H. Bahouri and J. Y. Chemin.Équations d'ondes quasilinéaires et effet dispersif. IMRN, vol. 21; (1999), pp. 1141-1178 | MR | Zbl

[Br] Y. Choquet Bruhat. Théorème d'existence pour certains systèmes d'équations aux dérivées partielles nonlinéaires., Acta Math. 88 (1952), 141-225. | MR | Zbl

[Ch-Kl] D. Christodoulou and S. Klainerman. The Global Nonlinear Stability of the Minkowski Space. Princeton Mathematical Series, 41. Princeton University Press, 1993 | MR | Zbl

[Ha-El] S. Hawking and G. Ellis. The large scale structure of spacetime. Cambridge Monographs on Mathematical Physics, 1973 | MR | Zbl

[H-K-M] Hughes, T. Kato and J. Marsden. Well posed quasilinear second order hyperbolic systems Arch. Rat. Mech. Anal. 63 (1976) no 3, 273-294. | MR | Zbl

[Kl1] S. Klainerman. A commuting vectorfield approach to Strichartz type inequalities and applications to quasilinear wave equations. IMRN, 2001, No 5, 221-274. | MR | Zbl

[Kl2] S. Klainerman. PDE as a unified subject Special Volume GAFA 2000, 279-315 XV-12 | MR | Zbl

[Kl-Ni] S. Klainerman and F. Nicolo. On the initial value problem in General Relativity. preprint

[Kl-Ro] S. Klainerman and I. Rodnianski. Improved local well posedness for quasilinear wave equations in dimension three. to appear in Duke Math. Journ. | MR | Zbl

[Kl-Ro1] S. Klainerman and I. Rodnianski. Rough solutions of the Einstein-vacuum equations. preprint | MR

[Kl-Ro2] S. Klainerman and I. Rodnianski. The causal structure of microlocalized, rough, Einstein metrics. preprint | MR

[Kl-Ro3] S. Klainerman and I. Rodnianski. Ricci defects of microlocalized, rough, Einstein metrics. preprint | MR

[Li] H. Lindblad, Counterexamples to local existence for semilinear wave equations. AJM, vol. 118; (1996), pp. 1-16 | MR | Zbl

[Po-Si] G. Ponce and T. Sideris. Local regularity of non linear wave equations in three space dimensions. CPDE, vol. 18; (1993), pp. 169-177 | MR | Zbl

[Sm] H. Smith. A parametrix construction for wave equations with C 1, 1 coefficients. Annales de L'Institut Fourier, vol. 48; (1998), pp. 797-835 | Numdam | MR | Zbl

[Sm-So] H. Smith and C. Sogge. On Strichartz and eigenfunction estimates for low regularity metrics. Math. Res. Lett., vol. 1; (1994), pp. 729-737 | MR | Zbl

[Sm-Ta1] H. Smith and D. Tataru. Sharp counterexamples for Strichartz estimates for low regularity metrics. Preprint | MR

[Sm-Ta2] H. Smith and D. Tataru. Sharp local well-posedness results for the nonlinear wave equation. Preprint

[Ta2] D. Tataru. Strichartz estimates for second order hyperbolic operators with non smooth coefficients. Preprint

[Ta1] D. Tataru. Strichartz estimates for operators with non smooth coefficients and the nonlinear wave equation.Amer. J. Math., vol. 122; (2000) | MR | Zbl

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