Centre de diffusion de revues académiques mathématiques

 
 
 
 

Journées équations aux dérivées partielles

Table des matières de ce volume | Article précédent | Article suivant
Peter Hintz; András Vasy
Quasilinear waves and trapping: Kerr-de Sitter space
Journées équations aux dérivées partielles (2014), Exp. No. 10, 15 p., doi: 10.5802/jedp.113
Article PDF

Résumé - Abstract

In these notes, we will describe recent work on globally solving quasilinear wave equations in the presence of trapped rays, on Kerr-de Sitter space, and obtaining the asymptotic behavior of solutions. For the associated linear problem without trapping, one would consider a global, non-elliptic, Fredholm framework; in the presence of trapping the same framework is available for spaces of growing functions only. In order to solve the quasilinear problem we thus combine these frameworks with the normally hyperbolic trapping results of Dyatlov and a Nash-Moser iteration scheme.

Bibliographie

[1] L. Andersson & P. Blue, “Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior”, Preprint, arXiv:1310.2664 (2013)
[2] A. Bachelot, “Gravitational scattering of electromagnetic field by Schwarzschild black-hole”, Ann. Inst. H. Poincaré Phys. Théor. 54 (1991) no. 3, p. 261-320 Numdam |  MR 1122656 |  Zbl 0743.53037
[3] A. Bachelot, Scattering of electromagnetic field by de Sitter-Schwarzschild black hole, Nonlinear hyperbolic equations and field theory (Lake Como, 1990), Pitman Res. Notes Math. Ser. 253, Longman Sci. Tech., Harlow, 1992, p. 23–35  MR 1175199 |  Zbl 0823.35162
[4] A. Sá Barreto & M. Zworski, “Distribution of resonances for spherical black holes”, Math. Res. Lett. 4 (1997) no. 1, p. 103-121  MR 1432814 |  Zbl 0883.35120
[5] M. Beals & M. Reed, “Microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems”, Trans. Amer. Math. Soc. 285 (1984) no. 1, p. 159-184 Article |  MR 748836 |  Zbl 0562.35093
[6] P. Blue & A. Soffer, “Phase space analysis on some black hole manifolds”, J. Funct. Anal. 256 (2009) no. 1, p. 1-90 Article |  MR 2475417 |  Zbl 1158.83007
[7] J.-F. Bony & D. Häfner, “Decay and non-decay of the local energy for the wave equation on the de Sitter-Schwarzschild metric”, Comm. Math. Phys. 282 (2008) no. 3, p. 697-719  MR 2426141 |  Zbl 1159.35007
[8] B. Carter, “Global structure of the Kerr family of gravitational fields”, Phys. Rev. 174 (1968), p. 1559-1571  Zbl 0167.56301
[9] B. Carter, “Hamilton-Jacobi and Schrödinger separable solutions of Einstein’s equations”, Comm. Math. Phys. 10 (1968), p. 280-310  MR 239841 |  Zbl 0162.59302
[10] M. Dafermos, G. Holzegel & I. Rodnianski, “A scattering theory construction of dynamical vacuum black holes”, Preprint, arxiv:1306.5364 (2013)
[11] M. Dafermos & I. Rodnianski, “A proof of Price’s law for the collapse of a self-gravitating scalar field”, Invent. Math. 162 (2005) no. 2, p. 381-457  MR 2199010 |  Zbl 1088.83008
[12] M. Dafermos & I. Rodnianski, “The wave equation on Schwarzschild-de Sitter space times”, Preprint, arXiv:07092766 (2007)
[13] M. Dafermos & I. Rodnianski, “The red-shift effect and radiation decay on black hole spacetimes”, Comm. Pure Appl. Math 62 (2009), p. 859-919  MR 2527808 |  Zbl 1169.83008
[14] M. Dafermos & I. Rodnianski, “Decay of solutions of the wave equation on Kerr exterior space-times I-II: The cases of $|a|\ll M$ or axisymmetry”, Preprint, arXiv:1010.5132 (2010)  MR 2730803
[15] M. Dafermos & I. Rodnianski, The black hole stability problem for linear scalar perturbations, in T. Damour et al, éd., Proceedings of the Twelfth Marcel Grossmann Meeting on General Relativity, World Scientific, Singapore, arXiv:1010.5137, 2011, p. 132-189
[16] M. Dafermos & I. Rodnianski, Lectures on black holes and linear waves, Evolution equations, Clay Math. Proc. 17, Amer. Math. Soc., Providence, RI, 2013, p. 97–205  MR 3098640
[17] M. Dafermos, I. Rodnianski & Y. Shlapentokh-Rothman, “Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case $|a| < M$”, Preprint, arXiv:1402.7034 (2014)
[18] R. Donninger, W. Schlag & A. Soffer, “A proof of Price’s law on Schwarzschild black hole manifolds for all angular momenta”, Adv. Math. 226 (2011) no. 1, p. 484-540 Article |  MR 2735767 |  Zbl 1205.83041
[19] S. Dyatlov, “Exponential energy decay for Kerr–de Sitter black holes beyond event horizons”, Math. Res. Lett. 18 (2011) no. 5, p. 1023-1035  MR 2875874 |  Zbl 1253.83020
[20] S. Dyatlov, “Quasi-normal modes and exponential energy decay for the Kerr-de Sitter black hole”, Comm. Math. Phys. 306 (2011) no. 1, p. 119-163 Article |  MR 2819421 |  Zbl 1223.83029
[21] S. Dyatlov, “Asymptotics of linear waves and resonances with applications to black holes”, Preprint, arXiv:1305.1723 (2013)
[22] S. Dyatlov, “Resonance projectors and asymptotics for $r$-normally hyperbolic trapped sets”, Preprint, arXiv:1301.5633 (2013)
[23] S. Dyatlov, “Spectral gaps for normally hyperbolic trapping”, Preprint, arXiv:1403.6401 (2013)
[24] S. Dyatlov & M. Zworski, “Trapping of waves and null geodesics for rotating black holes”, Phys. Rev. D 88 (2013)
[25] F. Finster, N. Kamran, J. Smoller & S.-T. Yau, “Decay of solutions of the wave equation in the Kerr geometry”, Comm. Math. Phys. 264 (2006) no. 2, p. 465-503 Article |  MR 2215614 |  Zbl 1194.83015
[26] F. Finster, N. Kamran, J. Smoller & S.-T. Yau, “Linear waves in the Kerr geometry: a mathematical voyage to black hole physics”, Bull. Amer. Math. Soc. (N.S.) 46 (2009) no. 4, p. 635-659 Article |  MR 2525736 |  Zbl 1177.83082
[27] P. Hintz, “Global well-posedness of quasilinear wave equations on asymptotically de Sitter spaces”, Preprint, arXiv:1311.6859 (2013)
[28] P. Hintz & A. Vasy, “Non-trapping estimates near normally hyperbolic trapping”, Preprint, arXiv:1306.4705 (2013)
[29] P. Hintz & A. Vasy, “Semilinear wave equations on asymptotically de Sitter, Kerr-de Sitter and Minkowski spacetimes”, Preprint, arXiv:1306.4705 (2013)
[30] P. Hintz & A. Vasy, “Global analysis of quasilinear wave equations on asymptotically Kerr-de Sitter spaces”, Preprint, arXiv:1404.1348 (2014)
[31] L. Hörmander, “On the existence and the regularity of solutions of linear pseudo-differential equations”, Enseignement Math. (2) 17 (1971), p. 99-163  MR 331124 |  Zbl 0224.35084
[32] L. Hörmander, The analysis of linear partial differential operators, vol. 1-4, Springer-Verlag, 1983  Zbl 0521.35002
[33] B. S. Kay & R. M. Wald, “Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation $2$-sphere”, Classical Quantum Gravity 4 (1987) no. 4, p. 893-898  MR 895907 |  Zbl 0647.53065
[34] J. Luk, “The null condition and global existence for nonlinear wave equations on slowly rotating Kerr spacetimes”, J. Eur. Math. Soc. (JEMS) 15 (2013) no. 5, p. 1629-1700 Article |  MR 3082240 |  Zbl 1280.35154
[35] J. Marzuola, J. Metcalfe, D. Tataru & M. Tohaneanu, “Strichartz estimates on Schwarzschild black hole backgrounds”, Comm. Math. Phys. 293 (2010) no. 1, p. 37-83 Article |  MR 2563798 |  Zbl 1202.35327
[36] R. B. Melrose, “Transformation of boundary problems”, Acta Math. 147 (1981) no. 3-4, p. 149-236  MR 639039 |  Zbl 0492.58023
[37] R. B. Melrose, The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics 4, A K Peters Ltd., Wellesley, MA, 1993  MR 1348401 |  Zbl 0796.58050
[38] R. B. Melrose, Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Marcel Dekker, 1994  MR 1291640 |  Zbl 0837.35107
[39] R. B. Melrose, A. Sá Barreto & A. Vasy, “Asymptotics of solutions of the wave equation on de Sitter-Schwarzschild space”, Comm. in PDEs 39 (2014) no. 3, p. 512-529  MR 3169793 |  Zbl 1286.35145
[40] S. Nonnenmacher & M. Zworski, “Decay of correlations for normally hyperbolic trapping”, Preprint, arXiv:1302.4483 (2013)
[41] X. Saint Raymond, “A simple Nash-Moser implicit function theorem”, Enseign. Math. (2) 35 (1989) no. 3-4, p. 217-226  MR 1039945 |  Zbl 0702.58011
[42] D. Tataru, “Local decay of waves on asymptotically flat stationary space-times”, Amer. J. Math. 135 (2013) no. 2, p. 361-401 Article |  MR 3038715 |  Zbl 1266.83033
[43] D. Tataru & M. Tohaneanu, “A local energy estimate on Kerr black hole backgrounds”, Int. Math. Res. Not. IMRN (2011) no. 2, p. 248-292 Article |  MR 2764864 |  Zbl 1209.83028
[44] M. Tohaneanu, “Strichartz estimates on Kerr black hole backgrounds”, Trans. Amer. Math. Soc. 364 (2012) no. 2, p. 689-702 Article |  MR 2846348 |  Zbl 1234.35275
[45] A. Vasy, Microlocal analysis of asymptotically hyperbolic spaces and high energy resolvent estimates, MSRI Publications, Cambridge University Press, 2012  MR 3135765
[46] A. Vasy, “Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces”, Inventiones Math. 194 (2013), p. 381-513, With an appendix by S. Dyatlov  MR 3117526
[47] R. M. Wald, “Note on the stability of the Schwarzschild metric”, J. Math. Phys. 20 (1979) no. 6, p. 1056-1058 Article |  MR 534342
[48] J. Wunsch & M. Zworski, “Resolvent estimates for normally hyperbolic trapped sets”, Ann. Henri Poincaré 12 (2011) no. 7, p. 1349-1385 Article |  MR 2846671 |  Zbl 1228.81170
[49] S. Yoshida, N. Uchikata & T. Futamase, “Quasinormal modes of Kerr-de Sitter black holes”, Phys. Rev. D 81 (2010) no. 4 Article |  MR 2659359
Copyright Cellule MathDoc 2018 | Crédit | Plan du site