While the physical properties of the gravitational field in the radiation regime are reasonably well understood, several mathematical questions remain unanswered. The question here is that of existence and properties of gravitational fields with asymptotic behavior compatible with existence of gravitational radiation. A framework to study those questions has been proposed by R. Penrose (R. Penrose, “Zero rest-mass fields including gravitation”, Proc. Roy. Soc. London A284 (1965), 159-203), and developed by H. Friedrich (H. Friedrich, “Cauchy problem for the conformal vacuum field equations in general relativity”, Commun. Math. Phys. 91 (1983), 445-472.), (H. Friedrich, “On the existence of n-geodesically complete or future complete solutions of Einstein's field equations with smooth asymptotic structure”, Commun. Math. Phys. 107 (1986), 587-609.), (-,“On the global existence and the asymptotic behavior of solutions to the Einstein-Maxwell-Yang-Mills equations”, Jour. Diff. Geom. 34 (1991), 275-345) using conformal completions techniques. In this conformal approach one has to 1) construct initial data, which satisfy the general relativistic constraint equations, with appropriate behavior near the conformal boundary, and 2) show a local (and perhaps also a global) existence theorem for the associated evolution problem. In this context solutions of the constraint equations can be found by solving a nonlinear elliptic system of equations, one of which resembles the Yamabe equation (and coincides with this equation in some cases), with the system degenerating near the conformal boundary. In the first part of the talk I (PTC) will describe the existence and boundary regularity results about this system obtained some years ago in collaboration with Helmut Friedrich and Lars Andersson. Some new applications of those techniques are also presented. In the second part of the talk I will describe some new results, obtained in collaboration with Olivier Lengard, concerning the evolution problem.
@article{JEDP_2000____A3_0,
author = {Chru\'sciel, Piotr T. and Lengard, Olivier},
title = {Polyhomogeneous solutions of wave equations in the radiation regime},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {3},
pages = {1--17},
publisher = {Universit\'e de Nantes},
year = {2000},
mrnumber = {2001h:35124},
zbl = {01808693},
language = {en},
url = {http://www.numdam.org/item/JEDP_2000____A3_0/}
}
TY - JOUR AU - Chruściel, Piotr T. AU - Lengard, Olivier TI - Polyhomogeneous solutions of wave equations in the radiation regime JO - Journées équations aux dérivées partielles PY - 2000 SP - 1 EP - 17 PB - Université de Nantes UR - http://www.numdam.org/item/JEDP_2000____A3_0/ LA - en ID - JEDP_2000____A3_0 ER -
Chruściel, Piotr T.; Lengard, Olivier. Polyhomogeneous solutions of wave equations in the radiation regime. Journées équations aux dérivées partielles (2000), article no. 3, 17 p. http://www.numdam.org/item/JEDP_2000____A3_0/
[1] and , On «hyperboloidal» Cauchy data for vacuum Einstein equations and obstructions to smoothness of Scri, Commun. Math. Phys. 161 (1994), 533-568. | MR | Zbl
[2] and , On asymptotic behaviour of solutions of the constraint equations in general relativity with «hyperboloidal boundary conditions», Dissert. Math. 355 (1996), 1-100. | MR | Zbl
[3] and , and , On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einsteins field equations, Comm. Math. Phys. 149 (1992), 587-612. | MR | Zbl
[4] and , Asymptotic behaviour of solutions and their derivatives, for semilinear elliptic problems with blowup on the boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), 155-171. | Numdam | MR | Zbl
[5] and , On second-order effects in the boundary behaviour of large solutions of semilinear elliptic problems, Diff. Integral Equations 11 (1998), 23-34. | MR | Zbl
[6] , Quasi-spherical metrics and prescribed scalar curvature, Jour. Diff. Geom. 37 (1993), 31-71. | MR | Zbl
[7] and , On the restricted validity of the thin sandwich conjecture, Phys. Rev. D 48 (1993), 3596-3599. | MR
[8] , , and , On asymptotically flat space-times with G2 invariant Cauchy surfaces, Annals of Phys. 237 (1995), 322-354, gr-qc/9404005. | MR | Zbl
[9] , , and , Gravitational waves in general relativity VII : Waves from axi-symmetric isolated systems, Proc. Roy. Soc. London A 269 (1962), 21-52. | MR | Zbl
[10] , Interaction des singularités pour les équations aux dérivées partielles non linéaires, Goulaouic-Meyer-Schwartz Seminar, 1981/1982, École Polytech., Palaiseau, 1982, pp. Exp. No. II, 12. | Numdam | Zbl
[11] , Global existence of wave maps, Proceedings of the IX International Conference on Waves and Stability in Continuous Media (Bari, 1997), vol. 1998, pp. 143-152. | MR | Zbl
[12] , Global existence theorems by the conformal method, Recent developments in hyperbolic equations (Pisa, 1987), Longman Sci. Tech., Harlow, 1988, pp. 16-37. | MR | Zbl
[13] , Global solutions of Yang-Mills equations on anti-de Sitter spacetime, Classical Quantum Gravity 6 (1989), 1781-1789. | MR | Zbl
[14] and , Existence globale d'applications harmoniques sur l'espace-temps de Minkowski M3, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), 167-170. | Zbl
[15] , , and , Solutions of constraints for Einstein equations, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 349-355. | MR | Zbl
[16] and , Solutions globales du système de Yang-Mills-Vlasov (masse nulle), C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 785-788. | MR | Zbl
[17] and , The Cauchy problem, General Relativity (A. Held, ed.), Plenum Press, New York, 1980, pp. 99-172. | MR
[18] , Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math. 39 (1986), 267-282. | MR | Zbl
[19] and , On the regularity of spherically symmetric wave maps, Commun. Pure Appl. Math 46 (1993), 1041-1091. | MR | Zbl
[20] and , On the asymptotic behavior of spherically symmetric wave maps, Duke Math. Jour. 71 (1993), 31-69. | MR | Zbl
[21] , Polyhomogeneous expansions at the boundary for some blowing-up solutions of a class of semi-linear elliptic equations, Tours preprint 136/96, ULR http://www.phys.univ-tours.fr/~piotr/papers/preprint136/ls.html, 1996.
[22] and , Solutions of wave equations in the radiating regime, in preparation.
[23] , , and , Gravitational waves in general relativity : XIV. Bondi expansions and the «polyhomogeneity» of Scri, Phil. Trans. Roy. Soc. A 350 (1995), 113-141. | MR | Zbl
[24] and , Vacuum spacetimes which are identically Schwarzschild near spatial infinity, talk given at the Santa Barbara Conference on Strong Gravitational Fields. June 22-26, 1999, http://dougpc.itp.ucsb.edu/online/gravity-c99/schoen/.
[25] , Cauchy problem for the conformal vacuum field equations in general relativity, Commun. Math. Phys. 91 (1983), 445-472. | MR | Zbl
[26] , Existence and structure of past asymptotically simple solutions of Einstein's field equations with positive cosmological constant, Jour. Geom. Phys. 3 (1986), 101-117. | MR | Zbl
[27] , On the existence of n-geodesically complete or future complete solutions of Einstein's field equations with smooth asymptotic structure, Commun. Math. Phys. 107 (1986), 587-609. | MR | Zbl
[28] , On the global existence and the asymptotic behavior of solutions to the Einstein - Maxwell - Yang-Mills equations, Jour. Diff. Geom. 34 (1991), 275-345. | MR | Zbl
[29] , Einstein equations and conformal structure : Existence of anti-de-Sitter-type space-times, Jour. Geom. and Phys. 17 (1995), 125-184. | MR | Zbl
[30] and , Conformal geodesics in general relativity, Proc. Roy. Soc. London Ser. A 414 (1987), 171-195. | MR | Zbl
[31] and , Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991), 186-225. | MR | Zbl
[32] and , A set of nonconstant mean curvature solutions of the Einstein constraint equations on closed manifolds, Classical Quantum Grav. 13 (1996), 1819-1847. | MR | Zbl
[33] and , Asymptotically hyperbolic non-constant mean curvature solutions of the Einstein constraint equations, Classical Quantum Grav. 14 (1997), A189-A201. | MR | Zbl
[34] , , and , Nonlinear hyperbolic smoothing at a focal point, preprint 12 on URL http://www.maths.univrennes1.fr/~metivier/preprints.html. | Zbl
[35] , A commutator proof of the propagation of polyhomogeneity for semi-linear equations, Commun. Partial Diff. Eq. 22 (1997), 435-463. | MR | Zbl
[36] , On the existence and convergence of polyhomogeneous expansions of zero-rest-mass fields, gr-qc/0005087 (2000). | Zbl
[37] , Polyhomogeneity and zero-rest-mass fields with applications to Newman-Penrose constants, Class. Quantum Grav. 17 (2000), no. 3, 605-621. | MR | Zbl
[38] , The gravitational field in the radiation regime, Ph.D. thesis, Université de Tours, in preparation.
[39] and , Interaction of nonlinear progressing waves for semi-linear wave equations, Ann. of Math. (2) 121 (1985), 187-213. | MR | Zbl
[40] , The global structure of simple space-times, Commun. Math. Phys. 123 (1989), 17-52. | MR | Zbl
[41] , Zero rest-mass fields including gravitation, Proc. Roy. Soc. London A284 (1965), 159-203. | MR | Zbl
[42] , Gravitational waves in general relativity VIII. Waves in asymptotically flat space-time, Proc. Roy. Soc. London A 270 (1962), 103-126. | MR | Zbl
[43] , King's College lecture notes on general relativity, May-June 1958, mimeographed notes; to be reprinted in Gen. Rel. Grav.
[44] , Radiation and boundary conditions in the theory of gravitation, Bull. Acad. Pol. Sci., Série sci. math., astr. et phys. VI (1958), 407-412. | MR | Zbl
[45] , Semilinear elliptic equations with uniform blow-up on the boundary, Jour. Anal. Math. 59 (1992), 231-250, Festschrift on the occasion of the 70th birthday of Shmuel Agmon. | MR | Zbl
[46] , General relativity, University of Chicago Press, Chicago, 1984. | MR | Zbl
