Polyhomogeneous solutions of wave equations in the radiation regime
Journées équations aux dérivées partielles (2000), article no. 3, 17 p.

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 Tadeusz and Lengard, Olivier},
title = {Polyhomogeneous solutions of wave equations in the radiation regime},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {3},
publisher = {Universit\'e de Nantes},
year = {2000},
zbl = {01808693},
mrnumber = {2001h:35124},
language = {en},
url = {http://www.numdam.org/item/JEDP_2000____A3_0/}
}
TY  - JOUR
AU  - Lengard, Olivier
TI  - Polyhomogeneous solutions of wave equations in the radiation regime
JO  - Journées équations aux dérivées partielles
PY  - 2000
DA  - 2000///
PB  - Université de Nantes
UR  - http://www.numdam.org/item/JEDP_2000____A3_0/
UR  - https://zbmath.org/?q=an%3A01808693
UR  - https://www.ams.org/mathscinet-getitem?mr=2001h:35124
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] L. Andersson and P.T. Chruściel, On «hyperboloidal» Cauchy data for vacuum Einstein equations and obstructions to smoothness of Scri, Commun. Math. Phys. 161 (1994), 533-568. | MR 95b:35220 | Zbl 0793.53084

[2] L. Andersson and P.T. Chruściel, On asymptotic behaviour of solutions of the constraint equations in general relativity with «hyperboloidal boundary conditions», Dissert. Math. 355 (1996), 1-100. | MR 97e:58217 | Zbl 0873.35101

[3] L. Andersson and P.T. Chruściel, and H. Friedrich, 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 93i:53040 | Zbl 0764.53027

[4] C. Bandle and M. Marcus, 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 96e:35038 | Zbl 0840.35033

[5] C. Bandle and M. Marcus, On second-order effects in the boundary behaviour of large solutions of semilinear elliptic problems, Diff. Integral Equations 11 (1998), 23-34. | MR 98m:35049 | Zbl 1042.35535

[6] R. Bartnik, Quasi-spherical metrics and prescribed scalar curvature, Jour. Diff. Geom. 37 (1993), 31-71. | MR 93i:53041 | Zbl 0786.53019

[7] R. Bartnik and G. Fodor, On the restricted validity of the thin sandwich conjecture, Phys. Rev. D 48 (1993), 3596-3599. | MR 94i:83009

[8] B. Berger, P.T. Chruściel, and V. Moncrief, On asymptotically flat space-times with G2 invariant Cauchy surfaces, Annals of Phys. 237 (1995), 322-354, gr-qc/9404005. | MR 95m:83005 | Zbl 0967.83507

[9] H. Bondi, M.G.J. Van Der Burg, and A.W.K. Metzner, Gravitational waves in general relativity VII : Waves from axi-symmetric isolated systems, Proc. Roy. Soc. London A 269 (1962), 21-52. | MR 26 #4793 | Zbl 0106.41903

[10] J.-M. Bony, 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 0498.35017

[11] Y. Choquet-Bruhat, 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 2000g:58043 | Zbl 0933.58028

[12] Y. Choquet-Bruhat, Global existence theorems by the conformal method, Recent developments in hyperbolic equations (Pisa, 1987), Longman Sci. Tech., Harlow, 1988, pp. 16-37. | MR 90g:58130 | Zbl 0734.35142

[13] Y. Choquet-Bruhat, Global solutions of Yang-Mills equations on anti-de Sitter spacetime, Classical Quantum Gravity 6 (1989), 1781-1789. | MR 90k:58032 | Zbl 0698.53040

[14] Y. Choquet-Bruhat and Chao Hao Gu, 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 0661.53043

[15] Y. Choquet-Bruhat, J. Isenberg, and V. Moncrief, Solutions of constraints for Einstein equations, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 349-355. | MR 93h:58151 | Zbl 0796.35161

[16] Y. Choquet-Bruhat and N. Noutchegueme, Solutions globales du système de Yang-Mills-Vlasov (masse nulle), C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 785-788. | MR 92g:58128 | Zbl 0715.53046

[17] Y. Choquet-Bruhat and J. York, The Cauchy problem, General Relativity (A. Held, ed.), Plenum Press, New York, 1980, pp. 99-172. | MR 82k:58028

[18] D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math. 39 (1986), 267-282. | MR 87c:35111 | Zbl 0612.35090

[19] D. Christodoulou and A. Tahvildar-Zadeh, On the regularity of spherically symmetric wave maps, Commun. Pure Appl. Math 46 (1993), 1041-1091. | MR 94e:58030 | Zbl 0744.58071

[20] D. Christodoulou and Z.S. Tahvildar-Zadeh, On the asymptotic behavior of spherically symmetric wave maps, Duke Math. Jour. 71 (1993), 31-69. | MR 94j:58044 | Zbl 0791.58105

[21] P.T. Chruściel, 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] P.T. Chruściel and O. Lengard, Solutions of wave equations in the radiating regime, in preparation.

[23] P.T. Chruściel, M.A.H. Maccallum, and D.B. Singleton, Gravitational waves in general relativity : XIV. Bondi expansions and the «polyhomogeneity» of Scri, Phil. Trans. Roy. Soc. A 350 (1995), 113-141. | MR 97f:83025 | Zbl 0829.53065

[24] J. Corvino and R. Schoen, 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] H. Friedrich, Cauchy problem for the conformal vacuum field equations in general relativity, Commun. Math. Phys. 91 (1983), 445-472. | MR 85g:83005 | Zbl 0555.35116

[26] H. Friedrich, 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 88c:83006 | Zbl 0592.53061

[27] 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. | MR 88b:83006 | Zbl 0659.53056

[28] H. Friedrich, 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 92i:58191 | Zbl 0737.53070

[29] H. Friedrich, Einstein equations and conformal structure : Existence of anti-de-Sitter-type space-times, Jour. Geom. and Phys. 17 (1995), 125-184. | MR 96j:83008 | Zbl 0840.53055

[30] H. Friedrich and B.G. Schmidt, Conformal geodesics in general relativity, Proc. Roy. Soc. London Ser. A 414 (1987), 171-195. | MR 89b:83041 | Zbl 0629.53063

[31] C.R. Graham and J.M. Lee, Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991), 186-225. | MR 92i:53041 | Zbl 0765.53034

[32] J. Isenberg and V. Moncrief, A set of nonconstant mean curvature solutions of the Einstein constraint equations on closed manifolds, Classical Quantum Grav. 13 (1996), 1819-1847. | MR 97h:83010 | Zbl 0860.53056

[33] J. Isenberg and J. Park, Asymptotically hyperbolic non-constant mean curvature solutions of the Einstein constraint equations, Classical Quantum Grav. 14 (1997), A189-A201. | MR 2000e:83009 | Zbl 0866.35126

[34] J.-L. Joly, G. Métivier, and J. Rauch, Nonlinear hyperbolic smoothing at a focal point, preprint 12 on URL http://www.maths.univrennes1.fr/~metivier/preprints.html. | Zbl 0989.35093

[35] M.S. Joshi, A commutator proof of the propagation of polyhomogeneity for semi-linear equations, Commun. Partial Diff. Eq. 22 (1997), 435-463. | MR 98c:35177 | Zbl 0876.35139

[36] J.A.V. Kroon, On the existence and convergence of polyhomogeneous expansions of zero-rest-mass fields, gr-qc/0005087 (2000). | Zbl 0973.83010

[37] J.A.V. Kroon, Polyhomogeneity and zero-rest-mass fields with applications to Newman-Penrose constants, Class. Quantum Grav. 17 (2000), no. 3, 605-621. | MR 2000m:83082 | Zbl 0947.83026

[38] O. Lengard, The gravitational field in the radiation regime, Ph.D. thesis, Université de Tours, in preparation.

[39] R. Melrose and N. Ritter, Interaction of nonlinear progressing waves for semi-linear wave equations, Ann. of Math. (2) 121 (1985), 187-213. | MR 86m:35005 | Zbl 0575.35063

[40] R.P.A.C. Newman, The global structure of simple space-times, Commun. Math. Phys. 123 (1989), 17-52. | MR 90i:83027 | Zbl 0683.53056

[41] R. Penrose, Zero rest-mass fields including gravitation, Proc. Roy. Soc. London A284 (1965), 159-203. | MR 30 #5774 | Zbl 0129.41202

[42] R. Sachs, Gravitational waves in general relativity VIII. Waves in asymptotically flat space-time, Proc. Roy. Soc. London A 270 (1962), 103-126. | MR 26 #7393 | Zbl 0101.43605

[43] A. Trautman, King's College lecture notes on general relativity, May-June 1958, mimeographed notes; to be reprinted in Gen. Rel. Grav.

[44] A. Trautman, Radiation and boundary conditions in the theory of gravitation, Bull. Acad. Pol. Sci., Série sci. math., astr. et phys. VI (1958), 407-412. | MR 20 #3736 | Zbl 0082.41201

[45] L. Véron, 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 94k:35113 | Zbl 0802.35042

[46] R.M. Wald, General relativity, University of Chicago Press, Chicago, 1984. | MR 86a:83001 | Zbl 0549.53001