×

Causal geometry of Einstein-vacuum spacetimes with finite curvature flux. (English) Zbl 1136.58018

The authors investigate how to circumvent a difficulty in the \(L^2\)- bounded curvature conjecture by showing that the geometry of null hypersurfaces of the vacuum Einstein equations can be controlled by using the initial data and the total curvature flux through a null outgoing hypersurface:
Contents include: Introduction; Null hypersurfaces; Motivation and second version of the Main Theorem; Bootstrap assumptions and preliminary estimates; Main estimates in Besov norms; Structure of error terms; Main estimates; and an Appendix on the fractional powers of the Laplace-Beltrami operator.

MSC:

58J45 Hyperbolic equations on manifolds
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35Q75 PDEs in connection with relativity and gravitational theory
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)

References:

[1] Bahouri, H., Chemin, J.Y.: Équations d?ondes quasilinéaires et estimation de Strichartz. Am. J. Math. 121, 1337-1777 (1999) · Zbl 0952.35073 · doi:10.1353/ajm.1999.0038
[2] Bahouri, H., Chemin, J.Y.: Équations d?ondes quasilinéaires et effet dispersif. Int. Math. Res. Not. 21, 1141-1178 (1999) · Zbl 0938.35106 · doi:10.1155/S107379289900063X
[3] Bony, J.M.: Calcul Symbolique et propagation des singularité pour les equations aux dérivées partielles nonlinéares. Ann. Sci. Éc. Norm. Supér., IV. Ser. 14, 209-256 (1981)
[4] Choquet Bruhat, Y.: Theoreme d?Existence pour certains systemes d?equations aux derivees partielles nonlineaires. Acta Math. 88, 141-225 (1952) · Zbl 0049.19201 · doi:10.1007/BF02392131
[5] Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of the Minkowski Space. Princeton Mathematical Series 41. Princeton, NJ: Princeton University Press 1993 · Zbl 0827.53055
[6] Hughes, T.J.R., Kato, T., Marsden, J.: Well posed quasilinear second order hyperbolic systems. Arch. Ration. Mech. Anal. 63, 273-294 (1976) · Zbl 0361.35046
[7] Friedrich, H., Stewart, J.M.: Characteristic initial data and wave front singulariries in general relativity. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 385, 345-371 (1983) · Zbl 0513.58043 · doi:10.1098/rspa.1983.0018
[8] Klainerman, S.: PDE as a unified subject Special Volume. Geom. Funct. Anal. 279-315 (2000) · Zbl 1002.35002
[9] Klainerman, S.: A commuting vectorfield approach to Strichartz type inequalities and applications to quasilinear wave equations. Int. Math. Res. Not. 221-274 (2001), no. 5 · Zbl 0993.35022
[10] Klainerman, S., Machedon, M.: Space-time estimates for null forms and the local existence theorem. CPAM 46, 1221-1268 (1993) · Zbl 0803.35095
[11] Klainerman, S., Nicolo, F.: The Evolution problem in General Relativity. Progr. Math. Phys. 25. Birkhäuser 2003
[12] Klainerman, S., Rodnianski, I.: Improved local well posedness for quasilinear wave equations in dimension three. Duke Math. J. 117, 1-124 (2003) · Zbl 1031.35091 · doi:10.1215/S0012-7094-03-11711-1
[13] Klainerman, S., Rodnianski, I.: Rough solutions of the Einstein vacuum equations. To appear in Ann. Math. · Zbl 1008.35079
[14] Klainerman, S., Rodnianski, I.: A geometric version of Litlewood-Paley theory. Submitted to Geom. Funct. Anal. · Zbl 1206.35080
[15] Klainerman, S., Rodnianski, I.: Sharp Trace Theorems on null hypersurfaces in Einstein backgrounds with finite curvature flux. Submitted to Geom. Funct. Anal. · Zbl 1206.35081
[16] Linblad, H.: Counterexamples to local existence for semilinear wave equations. Am. J. Math. 118, 1-16 (1996) · Zbl 0855.35080 · doi:10.1353/ajm.1996.0002
[17] Tataru, D.: Strichartz estimates for second order hyperbolic operators with non smooth coefficients. III. J. Am. Math. Soc. 15, 419-442 (2002) · Zbl 0990.35027 · doi:10.1090/S0894-0347-01-00375-7
[18] Smith, H., Tataru, D.: Sharp local well posedness results for the nonlinear wave equation. Submitted to Ann. Math. · Zbl 1098.35113
[19] Stein, E.: Topics in harmonic analysis related to Littlewood-Paley theory. Ann. Math. Stud. 63, 145 pages. Princeton University Press 1970 · Zbl 0193.10502
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.