The present article is a summary of the papers [10] and [11] which establish a bounded curvature theorem for the spacelike-characteristic Cauchy problem of general relativity. More precisely, we obtain a lower bound on the time of existence of classical solutions to the spacelike-characteristic Cauchy problem for Einstein equations in vacuum, depending only on the curvature fluxes through the initial spacelike and initial characteristic hypersurfaces and on suitable additional low regularity assumptions.
@article{SLSEDP_2019-2020____A2_0,
author = {Olivier Graf},
title = {Probl\`eme de {Cauchy} spatial-caract\'eristique avec courbure~$L^2$ en relativit\'e g\'en\'erale},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:3},
pages = {1--16},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2019-2020},
doi = {10.5802/slsedp.136},
zbl = {07436147},
language = {en},
url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.136/}
}
TY - JOUR AU - Olivier Graf TI - Problème de Cauchy spatial-caractéristique avec courbure $L^2$ en relativité générale JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:3 PY - 2019-2020 SP - 1 EP - 16 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.136/ DO - 10.5802/slsedp.136 LA - en ID - SLSEDP_2019-2020____A2_0 ER -
%0 Journal Article %A Olivier Graf %T Problème de Cauchy spatial-caractéristique avec courbure $L^2$ en relativité générale %J Séminaire Laurent Schwartz — EDP et applications %Z talk:3 %D 2019-2020 %P 1-16 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.136/ %R 10.5802/slsedp.136 %G en %F SLSEDP_2019-2020____A2_0
Olivier Graf. Problème de Cauchy spatial-caractéristique avec courbure $L^2$ en relativité générale. Séminaire Laurent Schwartz — EDP et applications (2019-2020), Talk no. 3, 16 p. doi : 10.5802/slsedp.136. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.136/
[1] S. Alexakis, A. Shao, On the geometry of null cones to infinity under curvature flux bounds, Class. Quantum Grav. 31 (2014), no. 19, 62 pp. | DOI | MR | Zbl
[2] S. Alexakis, A. Shao, Bounds on the Bondi energy by a flux of curvature, J. Eur. Math. Soc. 18 (2016), no. 9, 2045–2106. | DOI | MR | Zbl
[3] Y. Choquet-Bruhat, R. Geroch, Global aspects of the Cauchy problem in general relativity, Comm. Math. Phys. 14 (1969), no. 4, 329–335. | DOI | MR | Zbl
[4] D. Christodoulou, Bounded variation solutions of the spherically symmetric Einstein-scalar field equations, Commun. Pure Appl. Math. 46 (1993), no. 8, 1131–1220. | DOI | MR | Zbl
[5] D. Christodoulou, The instability of naked singularities in the gravitational collapse of a scalar field, Ann. of Math. (2) 149 (1999), no. 1, 183 pp. | DOI | MR | Zbl
[6] D. Christodoulou, The formation of black holes in general relativity, EMS Monogr. Math. (2009), x+589 pp. | DOI | Zbl
[7] D. Christodoulou, S. Klainerman, The global nonlinear stability of the Minkowski space, Princeton Univ. Press (1993), x+483 pp. | DOI | Zbl
[8] S. Czimek, An extension procedure for the constraint equations, Ann. PDE 4 (2018), no. 1, 130 pp. | DOI | MR | Zbl
[9] S. Czimek, Boundary harmonic coordinates on manifolds with boundary in low regularity, Comm. Math. Phys. 371 (2019), no. 3, 1131–1177. | DOI | MR | Zbl
[10] S. Czimek, O. Graf, The canonical foliation on null hypersurfaces in low regularity, (2019), 69 pp. | arXiv | DOI | MR
[11] S. Czimek, O. Graf, The spacelike-characteristic Cauchy problem of general relativity in low regularity, (2019), 91 pp. | arXiv | DOI
[12] Y. Fourès-Bruhat, Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires, Acta Math. 88 (1952), no. 1, 141–225. | DOI | Zbl
[13] S. Klainerman, M. Machedon, Finite energy solutions of the Yang-Mills equations in , Ann. of Math. (2) 142 (1995), no. 1, 39–119. | DOI | MR | Zbl
[14] S. Klainerman, F. Nicolò, On local and global aspects of the Cauchy problem in general relativity, Class. Quantum Grav. 16 (1999), no. 8, R73–R157. | DOI | MR | Zbl
[15] S. Klainerman, F. Nicolò, The evolution problem in general relativity, Prog. Math. Phys. Birkhäuser Boston Inc Boston MA 25 (2003), xiv+385 pp. | DOI | Zbl
[16] S. Klainerman, I. Rodnianski, Causal geometry of Einstein-vacuum spacetimes with finite curvature flux, Invent. Math. 159 (2005), no. 3, 437–529. | DOI | MR | Zbl
[17] S. Klainerman, I. Rodnianski, A geometric approach to the Littlewood–Paley theory, Geom. Funct. Anal. 16 (2006), no. 1, 126–163. | DOI | MR | Zbl
[18] S. Klainerman, I. Rodnianski, Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux, Geom. Funct. Anal. 16 (2006), no. 1, 164–229. | DOI | MR | Zbl
[19] S. Klainerman, I. Rodnianski, J. Szeftel, The bounded L2 curvature conjecture, Invent. Math. 202 (2015), no. 1, 91–216. | DOI | MR | Zbl
[20] J. Liu, J. Li, A robust proof of the instability of naked singularities of a scalar field in spherical symmetry, Commun. Math. Phys. (2018), 1–18. | DOI | MR | Zbl
[21] J. Luk, On the local existence for the characteristic initial value problem in general relativity, Int. Math. Res. Not. (2012), no. 20, 4625–4678. | DOI | MR | Zbl
[22] J. Luk, I. Rodnianski, Local propagation of impulsive gravitational waves, Comm. Pure Appl. Math. 68 (2015), no. 4, 511–624. | DOI | MR | Zbl
[23] F. Nicolò, Canonical foliation on a null hypersurface, J. Hyperbolic Differ. Equ. 1 (2004), no. 03, 367–428. | DOI | MR | Zbl
[24] S.-J. Oh, D. Tataru, The hyperbolic Yang–Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions, (2017), 145 pp. | arXiv | DOI | Zbl
[25] S.-J. Oh, D. Tataru, The threshold conjecture for the energy critical hyperbolic Yang–Mills equation, (2017), 45 pp. | arXiv | DOI | MR
[26] S.-J. Oh, D. Tataru, The Yang–Mills heat flow and the caloric gauge, (2017), 106 pp. | arXiv | DOI
[27] S.-J. Oh, D. Tataru, The hyperbolic Yang–Mills equation for connections in an arbitrary topological class, Comm. Math. Phys. 365 (2019), no. 2, 685–739. | DOI | MR | Zbl
[28] R. Penrose, Gravitational collapse: The role of general relativity, Riv. Nuovo Cimento (1969), no. I, 252–276. | DOI
[29] P. Petersen, Riemannian geometry, Grad. Texts in Math. Springer (2016), 499 pp. | DOI
[30] A. D. Rendall, Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations, Proc. Roy. Soc. London Ser. A 427 (1990), no. 1872, 221–239. | DOI | MR | Zbl
[31] R. K. Sachs, On the characteristic initial value problem in gravitational theory, J. Math. Phys. 3 (1962), no. 5, 908–914. | DOI | MR | Zbl
[32] A. Shao, New tensorial estimates in Besov spaces for time-dependent (2 + 1)-dimensional problems, J. Hyperbolic Differ. Equ 11 (2014), no. 04, 821–908. | DOI | MR | Zbl
[33] J. M. Stewart, H. Friedrich, Numerical relativity. I. The characteristic initial value problem, Proc. Roy. Soc. London Ser. A 384 (1982), no. 1787, 427–454. | DOI | MR | Zbl
[34] J. Szeftel, Parametrix for wave equations on a rough background I: Regularity of the phase at initial time, (2012), 145 pp. | arXiv | DOI | MR
[35] J. Szeftel, Parametrix for wave equations on a rough background II: Construction and control at initial time, (2012), 84 pp. | arXiv | DOI | MR
[36] J. Szeftel, Parametrix for wave equations on a rough background III: Space-time regularity of the phase, Astérisque (2018), no. 401, viii+321 pp. | DOI | Zbl
[37] J. Szeftel, Parametrix for wave equations on a rough background IV: Control of the error term, (2012), 284 pp. | arXiv | DOI | MR
[38] J. Szeftel, Sharp Strichartz estimates for the wave equation on a rough background, Ann. Sci. Ec. Norm. Supér. (4) 49 (2016), no. 6, 1279–1309. | DOI | MR | Zbl
[39] R. M. Wald, General relativity, Univ. Chicago Press (1984), 494 pp. | DOI | Zbl
[40] Q. Wang, On the geometry of null cones in Einstein-vacuum spacetimes, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 1, 285–328. | DOI | MR | Zbl
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