Abstract
One of the central difficulties of settling the L2-bounded curvature conjecture for the Einstein-Vacuum equations is to be able to control the causal structure of spacetimes with such limited regularity. In this paper we show how to circumvent this difficulty by showing that the geometry of null hypersurfaces of Enstein-Vacuum spacetimes can be controlled in terms of initial data and the total curvature flux through the hypersurface.
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Bahouri, H., Chemin, J.Y.: Équations d’ondes quasilinéaires et estimation de Strichartz. Am. J. Math. 121, 1337–1777 (1999)
Bahouri, H., Chemin, J.Y.: Équations d’ondes quasilinéaires et effet dispersif. Int. Math. Res. Not. 21, 1141–1178 (1999)
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)
Choquet Bruhat, Y.: Theoreme d’Existence pour certains systemes d’equations aux derivees partielles nonlineaires. Acta Math. 88, 141–225 (1952)
Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of the Minkowski Space. Princeton Mathematical Series 41. Princeton, NJ: Princeton University Press 1993
Hughes, T.J.R., Kato, T., Marsden, J.: Well posed quasilinear second order hyperbolic systems. Arch. Ration. Mech. Anal. 63, 273–294 (1976)
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)
Klainerman, S.: PDE as a unified subject Special Volume. Geom. Funct. Anal. 279–315 (2000)
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
Klainerman, S., Machedon, M.: Space-time estimates for null forms and the local existence theorem. CPAM 46, 1221–1268 (1993)
Klainerman, S., Nicolo, F.: The Evolution problem in General Relativity. Progr. Math. Phys. 25. Birkhäuser 2003
Klainerman, S., Rodnianski, I.: Improved local well posedness for quasilinear wave equations in dimension three. Duke Math. J. 117, 1–124 (2003)
Klainerman, S., Rodnianski, I.: Rough solutions of the Einstein vacuum equations. To appear in Ann. Math.
Klainerman, S., Rodnianski, I.: A geometric version of Litlewood-Paley theory. Submitted to Geom. Funct. Anal.
Klainerman, S., Rodnianski, I.: Sharp Trace Theorems on null hypersurfaces in Einstein backgrounds with finite curvature flux. Submitted to Geom. Funct. Anal.
Linblad, H.: Counterexamples to local existence for semilinear wave equations. Am. J. Math. 118, 1–16 (1996)
Tataru, D.: Strichartz estimates for second order hyperbolic operators with non smooth coefficients. III. J. Am. Math. Soc. 15, 419–442 (2002)
Smith, H., Tataru, D.: Sharp local well posedness results for the nonlinear wave equation. Submitted to Ann. Math.
Stein, E.: Topics in harmonic analysis related to Littlewood-Paley theory. Ann. Math. Stud. 63, 145 pages. Princeton University Press 1970
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Mathematics Subject Classification (1991)
35J10
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Klainerman, S., Rodnianski, I. Causal geometry of Einstein-Vacuum spacetimes with finite curvature flux. Invent. math. 159, 437–529 (2005). https://doi.org/10.1007/s00222-004-0365-4
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DOI: https://doi.org/10.1007/s00222-004-0365-4