[1] Y. Choquet-Bruhat Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires, Acta Math., Volume 88 (1952), pp. 141-225 | DOI

[2] Y. Choquet-Bruhat General Relativity and the Einstein Equations, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2009

[3] G. A. Burnett The high-frequency limit in general relativity, J. Math. Phys., Volume 30 (1989) no. 1, pp. 90-96 | DOI | Zbl

[4] Y. Choquet-Bruhat Construction de solutions radiatives approchées des équations d’Einstein, Commun. Math. Phys., Volume 12 (1969), pp. 16-35 http://projecteuclid.org/euclid.cmp/1103841306 | DOI

[5] D. R. Brill; J. B. Hartle Method of the self-consistent field in general relativity and its application to the gravitational geon, Phys. Rev., Volume 135 (1964), p. B271-B278 | DOI | Zbl

[6] R. A. Isaacson Gravitational radiation in the limit of high frequency. I. The linear approximation and geometrical optics, Phys. Rev., Volume 166 (1968), pp. 1263-1271 | DOI

[7] R. A. Isaacson Gravitational radiation in the limit of high frequency. II. Nonlinear terms and the effective stress tensor, Phys. Rev., Volume 166 (1968), pp. 1272-1279 | DOI

[8] M. A. H. MacCallum; A. H. Taub The averaged Lagrangian and high-frequency gravitational waves, Commun. Math. Phys., Volume 30 (1973) no. 2, pp. 153-169 | DOI

[9] S. R. Green; R. M. Wald New framework for analyzing the effects of small scale inhomogeneities in cosmology, Phys. Rev. D, Volume 83 (2011) no. 8, 084020 | DOI

[10] Y. Choquet-Bruhat Une Mathématicienne dans Cet étrange Univers : Mémoires, Odile Jacob, Paris, 2016

[11] C. Huneau; J. Luk High-frequency solutions to the Einstein equations, Class. Quantum Gravity, Volume 41 (2024) no. 14, 143002 | DOI | Zbl

[12] A. Einstein Näherungsweise integration der feldgleichungen der gravitation, Sitzungsber. Kgl. Preuss. Akad. Wiss., Volume 1916 (1916), pp. 688-696 | Zbl

[13] C. W. Misner; K. S. Thorne; J. A. Wheeler Gravitation, W. H. Freeman and Co., San Francisco, CA, 1973

[14] Y. Choquet-Bruhat Ondes asymptotiques pour un système d’équations aux dérivées partielles non linéaires, C. R. Acad. Sci. Paris Sér. A, Volume 264 (1967), pp. 625-628 | Zbl

[15] Y. Choquet-Bruhat Ondes asymptotiques et approchées pour des systèmes d’équations aux dérivées partielles non linéaires, J. Math. Pures Appl. (9), Volume 48 (1969), pp. 117-158 | Zbl

[16] Y. Choquet-Bruhat Ondes asymptotiques et approchées pour un système d’équations aux dérivées partielles non linéaires, C. R. Acad. Sci., Volume 264 (1967), pp. 625-638

[17] Y. Choquet-Bruhat Ondes asymptotiques et approchées pour un système d’équations aux dérivées partielles non linéaires, Sémin. Jean Leray, Volume 3 (1969), pp. 1-10 (MR:255964. Zbl:0177.36404) | Zbl

[18] A. M. Anile Relativistic Fluids and Magneto-fluids: With Applications in Astrophysics and Plasma Physics, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1990 | DOI

[19] Y. Choquet-Bruhat Approximate radiative solutions of Einstein–Maxwell equations, Relativity and Gravitation (G. Kuper Charles, ed.), Gordon and Breach Science Publishers, Inc, New York, 1971, pp. 81-86 https://www.osti.gov/biblio/4661970

[20] Y. Choquet-Bruhat; A. Greco Ondes gravitationnelles à haute fréquence et interaction avec la matière, C. R. Acad. Sci., Paris, Sér. II, Fasc. b, Volume 323 (1996) no. 2, pp. 117-124 | Zbl

[21] Y. Choquet-Bruhat; A. H. Taub High-frequency, self-gravitating, charged scalar fields, Gen. Relativ. Gravit., Volume 8 (1977) no. 8, pp. 561-571 | DOI

[22] Y. Choquet-Bruhat; A. Greco High frequency asymptotic solutions of Yang–Mills and associated fields, J. Math. Phys., Volume 24 (1983) no. 2, pp. 377-379 | DOI | Zbl

[23] Y. Choquet-Bruhat Ondes à haute fréquence pour la gravitation avec termes de Gauss–Bonnet, C. R. Acad. Sci. Paris Sér. I Math., Volume 307 (1988) no. 12, pp. 693-696 | Zbl

[24] Y. Choquet-Bruhat High frequency waves for stringy gravity, Proceedings of the Fifth Marcel Grossmann Meeting on General Relativity, Part A, B (Teaneck, NJ) (D. G. Blair; M. J. Buckingham; R. Ruffini, eds.), World Scientific Publishing, Singapore, 1989, pp. 349-361 (Perth, 1988). MR:1056882

[25] H. Andréasson The Einstein–Vlasov system/kinetic theory, Living Rev. Relativ., Volume 14 (2011), 4 | DOI | Zbl

[26] Y. Choquet-Bruhat Problème de Cauchy pour le système intégro-différentiel d’Einstein-Liouville. (Cauchy problem for the Einstein-Liouville integro-differential system), Ann. Inst. Fourier, Volume 21 (1971) no. 3, pp. 181-201 https://eudml.org/doc/74046 | DOI | Zbl

[27] A. D. Rendall The Newtonian limit for asymptotically flat solutions of the Vlasov–Einstein system, Commun. Math. Phys., Volume 163 (1994) no. 1, pp. 89-112 | DOI | Zbl

[28] B. Le Floch; P. G. Lefloch Compensated compactness and corrector stress tensor for the Einstein equations in ${𝕋}^2$ symmetry, Port. Math. (N.S.), Volume 77 (2020) no. 3–4, pp. 409-421 | DOI | Zbl

[29] L. Tartar The General Theory of Homogenization, Lecture Notes of the Unione Matematica Italiana, 7, Springer-Verlag, Berlin; UMI, Bologna, 2009 | DOI

[30] D. Christodoulou Global solutions of nonlinear hyperbolic equations for small initial data, Commun. Pure Appl. Math., Volume 39 (1986) no. 2, pp. 267-282 | DOI | Zbl

[31] S. Klainerman The null condition and global existence to nonlinear wave equations, Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1 (Santa Fe, NM, 1984) (Lectures in Applied Mathematics), Volume 23, American Mathematical Society, Providence, RI, 1986, pp. 293-326 | Zbl

[32] F. Murat Compacite par compensation, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., Volume 5 (1978), pp. 489-507 https://eudml.org/doc/83787 | Zbl

[33] L. Tartar Compensated compactness and applications to partial differential equations, Nonlinear Analysis and Mechanics: Heriot–Watt Symposium, Vol. 4, Edinburgh 1979 (Research Notes in Mathematics), Volume 39, Pitman Publishing Ltd., London, 1979, pp. 136-212 | Zbl

[34] C. De Lellis; L. J. Székelyhidi Dissipative continuous Euler flows, Invent. Math., Volume 193 (2013) no. 2, pp. 377-407 | DOI | Zbl

[35] C. De Lellis; L. J. Székelyhidi Weak stability and closure in turbulence, Phil. Trans. R. Soc. A (2022) no. 380, 20210091 | DOI

[36] P. Isett A proof of Onsager’s conjecture, Ann. Math. (2), Volume 188 (2018) no. 3, pp. 871-963 | DOI | Zbl

[37] J. Shatah Normal forms and quadratic nonlinear Klein-Gordon equations, Commun. Pure Appl. Math., Volume 38 (1985), pp. 685-696 | DOI | Zbl

[38] J.-L. Joly; G. Metivier; J. Rauch Transparent nonlinear geometric optics and Maxwell-Bloch equations, J. Differ. Equ., Volume 166 (2000) no. 1, pp. 175-250 | DOI | Zbl

[39] D. Lannes Space time resonances [after Germain, Masmoudi, Shatah], Séminaire Bourbaki Volume 2011/2012 exposés 1043-1058 (Astérisque, no. 352), Société mathématique de France, 2013 | Zbl

[40] D. Christodoulou; S. Klainerman The Global Nonlinear Stability of the Minkowski Space, Princeton Mathematical Series, 41, Princeton University Press, Princeton, NJ, 1993

[41] H. Lindblad; I. Rodnianski The global stability of Minkowski space–time in harmonic gauge, Ann. of Math. (2), Volume 171 (2010) no. 3, pp. 1401-1477 | DOI | Zbl

[42] Y. Choquet-Bruhat The null condition and asymptotic expansions for the Einstein equations, Ann. Phys. (8), Volume 9 (2000) no. 3–5, pp. 258-266 | DOI | Zbl

[43] H. Lindblad; I. Rodnianski The weak null condition for Einstein’s equations, C. R., Math., Acad. Sci. Paris, Volume 336 (2003) no. 11, pp. 901-906 | DOI | Zbl

[44] S. Klainerman; I. Rodnianski; J. Szeftel The bounded L${}^2$ curvature conjecture, Invent. Math., Volume 202 (2015) no. 1, pp. 91-216 | DOI | Zbl

[45] T. Buchert Dark energy from structure: a status report, Gen. Relativ. Gravit., Volume 40 (2008) no. 2, pp. 467-527 | DOI | Zbl

[46] E. W. Kolb; S. Matarrese; A. Riotto On cosmic acceleration without dark energy, New J. Phys., Volume 8 (2006) no. 12, 322 | DOI

[47] S. R. Green; R. M. Wald Examples of backreaction of small-scale inhomogeneities in cosmology, Phys. Rev. D, Volume 87 (2013), 124037 | DOI | Zbl

[48] T. Buchert; M. Carfora; G. F. R. Ellis; E. W. Kolb; M. A. H. MacCallum; J. J. Ostrowski; S. Räsänen; B. F. Roukema; L. Andersson; A. A. Coley; D. L. Wiltshire Is there proof that backreaction of inhomogeneities is irrelevant in cosmology?, Class. Quantum Gravity, Volume 32 (2015) no. 21, 215021 | DOI | Zbl

[49] S. R. Green; R. M. Wald Comments on backreaction, preprint, 2015 | arXiv

[50] A. Guerra; R. T. da Costa Oscillations in wave map systems and homogenization of the Einstein equations in symmetry, preprint, 2021 | arXiv

[51] C. Huneau; J. Luk High-frequency backreaction for the Einstein equations under polarized $𝕌\left(1\right)$-symmetry, Duke Math. J., Volume 167 (2018) no. 18, pp. 3315-3402 | DOI | Zbl

[52] C. Huneau; J. Luk Trilinear compensated compactness and Burnett’s conjecture in general relativity, Ann. Sci. Éc. Norm. Supér. (4), Volume 57 (2024) no. 2, pp. 385-472 | DOI | Zbl

[53] C. Huneau; J. Luk Burnett’s conjecture in generalized wave coordinates, preprint, 2024 | arXiv

[54] C. Huneau; J. Luk High-frequency backreaction for the Einstein equations under $𝕌\left(1\right)$ symmetry: from Einstein-dust to Einstein–Vlasov, 2024 (In preparation)

[55] J. Luk; I. Rodnianski High-frequency limits and null dust shell solutions in general relativity, preprint, 2020 | arXiv

[56] A. Touati Geometric optics approximation for the einstein vacuum equations, Commun. Math. Phys., Volume 402 (2023) no. 3, pp. 3109-3200 | DOI | Zbl

[57] A. Touati The reverse Burnett conjecture for null dusts, preprint, 2024 | arXiv

[58] D. Christodoulou Bounded variation solutions of the spherically symmetric Einstein-scalar field equations, Commun. Pure Appl. Math., Volume 46 (1993) no. 8, pp. 1131-1220 | DOI

[59] G. Métivier The mathematics of nonlinear optics, Handbook of Differential Equations (C. M. Dafermos; M. Pokorný, eds.) (Handbook of Differential Equations: Evolutionary Equations), Volume 5, North-Holland, Amsterdam, 2009, pp. 169-313 | DOI | Zbl

[60] J. Rauch Hyperbolic Partial Differential Equations and Geometric Optics, Graduate Studies in Mathematics, 133, American Mathematical Society, Providence, RI, 2012 | DOI

[61] P. D. Lax Asymptotic solutions of oscillatory initial value problems, Duke Math. J., Volume 24 (1957), pp. 627-646 | DOI | Zbl

[62] L. Garding; T. Kotake; J. Leray Uniformisation et développement asymptotique de la solution du problème de Cauchy linéaire, à données holomorphes; analogie avec la théorie des ondes asymptotiques et approchées. (Problème de Cauchy I bis et VI), Bull. Soc. Math. Fr., Volume 92 (1964), pp. 263-361 | DOI | Zbl

[63] P.-Y. Jeanne Geometric optics for gauge invariant semilinear systems, Mém. Soc. Math. Fr., Nouv. Sér., Volume 90 (2002), p. vi + 160 https://smf.emath.fr/publications/optique-geometrique-pour-des-systemes-semi-lineaires-avec-invariance-de-jauge | Zbl

[64] T. Salvi Multi-phase high frequency solutions to Klein–Gordon–Maxwell equations in Lorenz gauge in (3 + 1) Minkowski spacetime, preprint, 2024 | arXiv

[65] J. K. Hunter; J. B. Keller Weakly nonlinear high frequency waves, Commun. Pure Appl. Math., Volume 36 (1983), pp. 547-569 | DOI | Zbl

[66] J.-L. Joly; G. Metivier; J. Rauch Generic rigorous asymptotic expansions for weakly nonlinear multidimensional oscillatory waves, Duke Math. J., Volume 70 (1993) no. 2, pp. 373-404 | DOI | Zbl

[67] Y. Choquet-Bruhat; V. Moncrief An existence theorem for the reduced Einstein equation, C. R. Acad. Sci. Paris Sér. I Math., Volume 319 (1994) no. 2, pp. 153-159 (MR: 1288395) | Zbl

[68] Y. Choquet-Bruhat; V. Moncrief Future global in time Einsteinian spacetimes with U(1) isometry group, Ann. Henri Poincaré, Volume 2 (2001) no. 6, pp. 1007-1064 | DOI | Zbl

[69] C. Huneau Constraint equations for 3 + 1 vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case. II, Asymp. Analy., Volume 96 (2016) no. 1, pp. 51-89 | DOI

[70] C. Huneau Stability of Minkowski space–time with a translation space-like Killing field, Ann. PDE, Volume 4 (2018) no. 1, 12 | DOI

[71] S. Alexakis; N. T. Carruth Squeezing a fixed amount of gravitational energy to arbitrarily small scales, in $𝕌\left(1\right)$ symmetry, preprint, 2022 | arXiv

[72] J. Luk; M. Van de Moortel Nonlinear interaction of three impulsive gravitational waves I: main result and the geometric estimates, preprint, 2021 ([gr-qc]) | arXiv

[73] J. Luk; M. Van de Moortel Nonlinear interaction of three impulsive gravitational waves. II: The wave estimates, Ann. PDE, Volume 9 (2023) no. 1, 10 | DOI | Zbl

[74] C. Huneau; J. Luk Einstein equations under polarized $𝕌\left(1\right)$ symmetry in an elliptic gauge, Commun. Math. Phys., Volume 361 (2018) no. 3, pp. 873-949 | DOI | Zbl

[75] A. Touati Einstein vacuum equations with $𝕌\left(1\right)$ symmetry in an elliptic gauge: Local well-posedness and blow-up criterium, J. Hyperbolic Differ. Equ., Volume 19 (2022) no. 04, pp. 635-715 | DOI | Zbl

[76] A. J. Fang Nonlinear stability of the slowly-rotating Kerr-de Sitter family, preprint, 2021 | arXiv

[77] A. J. Fang Linear stability of the slowly-rotating Kerr-de Sitter family, preprint, 2022 | arXiv

[78] P. Hintz; A. Vasy The global non-linear stability of the Kerr-de Sitter family of black holes, Acta Math., Volume 220 (2018) no. 1, pp. 1-206 | DOI

[79] F. Pretorius Evolution of binary black-hole spacetimes, Phys. Rev. Lett., Volume 95 (2005), 121101 | DOI

[80] Y. Choquet-Bruhat; H. Friedrich Motion of isolated bodies, Class. Quantum Gravity, Volume 23 (2006) no. 20, pp. 5941-5949 | DOI

[81] A. Touati High-frequency solutions to the constraint equations, Commun. Math. Phys., Volume 402 (2023) no. 1, pp. 97-140 | DOI | Zbl

[82] N. Burq Semi-classical measures and defect measures, Séminaire Bourbaki. Volume 1996/97. Exposés 820–834, Société Mathématique de France, Paris, 1997, pp. 167-195 ex (French) | Zbl

[83] P. Gérard Microlocal defect measures, Commun. Partial Differ. Equ., Volume 16 (1991) no. 11, pp. 1761-1794 | DOI | Zbl

[84] L. Tartar H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. R. Soc. Edinb., Sect. A, Math., Volume 115 (1990) no. 3–4, pp. 193-230 | DOI | Zbl

[85] G. A. Francfort; F. Murat Oscillations and energy densities in the wave equation, Commun. Partial Differ. Equ., Volume 17 (1992) no. 11–12, pp. 1785-1865 | DOI | Zbl

[86] C. Bardos; G. Lebeau; J. Rauch Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., Volume 30 (1992) no. 5, pp. 1024-1065 | DOI | Zbl

[87] N. Burq; P. Gérard A necessary and sufficient condition for the exact controllability of the wave equation, C. R. Acad. Sci., Paris, Sér. I, Math., Volume 325 (1997) no. 7, pp. 749-752 | DOI | Zbl

[88] J. L. Joly; G. Métivier; J. Rauch Trilinear compensated compactness and nonlinear geometric optics, Ann. Math. (2), Volume 142 (1995) no. 1, pp. 121-169 | DOI | Zbl

[89] A. D. Ionescu; B. Pausader The Einstein–Klein–Gordon coupled system: global stability of the Minkowski solution, Annals of Mathematics Studies, 213, Princeton University Press, Princeton, NJ, 2022

[90] D. Christodoulou The Formation of Black Holes in General Relativity, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2009 | DOI

[91] M. Dafermos; G. Holzegel; I. Rodnianski The linear stability of the Schwarzschild solution to gravitational perturbations, Acta Math., Volume 222 (2019) no. 1, pp. 1-214 | DOI | Zbl

[92] S. Klainerman; F. Nicolò The Evolution Problem in General Relativity, Progress in Mathematical Physics, 25, Birkhäuser Boston, Inc., Boston, MA, 2003 | DOI

[93] J. Luk On the local existence for the characteristic initial value problem in general relativity, Int. Math. Res. Not., Volume 2012 (2012) no. 20, pp. 4625-4678 | DOI | Zbl

[94] J. Luk; I. Rodnianski Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations, Camb. J. Math., Volume 5 (2017) no. 4, pp. 435-570 | DOI

[95] J. Luk; I. Rodnianski Local propagation of impulsive gravitational waves, Commun. Pure Appl. Math., Volume 68 (2015) no. 4, pp. 511-624 | DOI

[96] S. W. Hawking Gravitational radiation from collapsing cosmic string loops, Phys. Lett. B, Volume 246 (1990) no. 1, pp. 36-38 | DOI

[97] R. Penrose Naked singularities, 6th Texas symposium on Relativistic astrophysics. New York, NY, USA, December 18–22, 1972, New York Academy of Sciences, New York, 1973, pp. 125-134 | Zbl

[98] L. Bigorgne; D. Fajman; J. Joudioux; J. Smulevici; M. Thaller Asymptotic stability of Minkowski space–time with non-compactly supported massless Vlasov matter, Arch. Ration. Mech. Anal., Volume 242 (2021) no. 1, pp. 1-147 | DOI

[99] M. Taylor The global nonlinear stability of Minkowski space for the massless Einstein–Vlasov system, Ann. PDE, Volume 3 (2017) no. 1, 9 | DOI

[100] H. Andréasson; D. Fajman; M. Thaller Models for self-gravitating photon shells and geons, Ann. Henri Poincaré, Volume 18 (2017) no. 2, pp. 681-705 | DOI

[101] G. Lebeau Nonlinear optics and supercritical wave equation, Bull. Soc. R. Sci. Liège, Volume 70 (2001) no. 4–6, pp. 267-306 | Zbl

[102] G. Alì; J. K. Hunter Large amplitude gravitational waves, J. Math. Phys., Volume 40 (1999) no. 6, pp. 3035-3052 | DOI

[103] G. Alí; J. K. Hunter Diffractive nonlinear geometrical optics for variational wave equations and the Einstein equations, Commun. Pure Appl. Math., Volume 60 (2007) no. 10, pp. 1522-1557 | DOI

[104] S. Klainerman; I. Rodnianski On the formation of trapped surfaces, Acta Math., Volume 208 (2012) no. 2, pp. 211-333 | DOI

[105] X. An; J. Luk Trapped surfaces in vacuum arising dynamically from mild incoming radiation, Adv. Theor. Math. Phys., Volume 21 (2017) no. 1, pp. 1-120 | DOI

[106] A. Abrahams; A. Anderson; Y. Choquet-Bruhat; J. W. jun York Un système hyperbolique non strict pour les équations d’Einstein, C. R. Acad. Sci., Paris, Sér. II, Fasc. b, Volume 323 (1996) no. 12, pp. 835-841

[107] J. Leray; Y. Ohya Équations et systèmes non-linéaires, hyperboliques non-stricts, Math. Ann., Volume 170 (1967), pp. 167-205 https://eudml.org/doc/161535 | DOI

[108] Y. Choquet-Bruhat Nonstrict and strict hyperbolic systems for the Einstein equations, preprint, 2001 | arXiv